TOPOLOGICAL EQUIVALENCE OF LINEAR

REPRESENTATIONS FOR CYCLIC GROUPS

Ian Hambleton and Erik K. Pedersen

June 17, 1997

Abstract. In this paper we prove that the Reidemeister torsion invariants determine topological equivalence of G–representations, for G a finite cyclic group.

1. Introduction

Let G be a finite group and V, V′ finite dimensional real orthogonal representations of G. Then V is said to be topologically equivalent to V′ (denoted V ∼_{t} V′) if there exists a homeomorphism h: V → V′ which is G-equivariant. If V, V′ are topologically equivalent, but not linearly isomorphic, then such a homeomorphism is called a non-linear similarity. These notions were introduced and studied by de Rham [33], [34], and developed extensively in [3], [4], [23], [24], and [7]. In this paper, we complete de Rham’s program by showing that Reidemeister torsion invariants and number theory determine non-linear similarity for finite cyclic groups (see Section 2 for the detailed statement of results in Theorems A–D).

In [3], Cappell and Shaneson showed that non-linear similarities exist for cyclic groups G = C(4q) of every order 4q ≧ 8. On the other hand, if G = C(q) or G = C(2q), for q odd, Hsiang–Pardon [23] and Madsen–Rothenberg [24] proved that topological equivalence of G–representations implies linear equivalence (the case G = C(4) is trivial). Since linear G-equivalence for general finite groups G is detected by restriction to cyclic subgroups, it is reasonable to study this case first. For the rest of the paper, unless otherwise mentioned, G denotes a finite cyclic group.

Further positive results can be obtained by imposing assumptions on the isotropy subgroups allowed in V_{1} and V_{2}. For example, if the V_{i} are free G-representions, then de Rham [33] proved in 1935 that piecewise linear similarity implies linear equivalence, by using Reidemeister torsion and the Franz Independence Lemma. Topological invariance of Whitehead torsion shows that his method also rules out non–linear similarity in this case. In [19,Thm.A] we studied “first–time” similarities, where Res_{K} V_{1} ≅ Res_{K} V_{2} for all proper subgroups K ⫋ G, and showed that

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Partially supported by NSERC grant A4000 and NSF grant DMS 9104026. The authors also wish to thank the Max Planck Institut für Mathematik, Bonn, for its hospitality and support.

topological equivalence implies linear equivalence if V_{1}, V_{2} have no isotropy subgroup of index 2. This result is an application of bounded surgery theory (see [18], [19,§4]), and these techniques are extended here to provide a necessary and sufficient condition for non-linear similarity in terms of the vanishing of a bounded transfer map (see Theorem 3.5). This provides a new approach to de Rham’s problem. The main work of the present paper is to establish methods for effective calculation of the bounded transfer.

In [4], Cappell and Shaneson initiated the study of stable topological equvalence for G–representations. We say that V_{1} and V_{2} are stably topologically similar (V_{1} ≈_{t} V_{2}) if there exists a G–representation W such that V_{1} ⊕ W ∼_{t} V_{2} ⊕ W. Let R_{Top}(G) = R(G)/R_{t}(G) denote the quotient group of the real representation ring of G by the subgroup R_{t}(G) = {[V_{1}]−[V_{2}] | V_{1} ≈_{t} V_{2}}. In [4], R_{Top}(G)⊗Z[1/2] was computed, and the torsion subgroup was shown to be 2–primary. As an application of our general results, we determine the structure of the torsion in R_{Top}(G), for G cyclic (see Section 20).

One interesting question in non–linear similarity concerns the minimum possible dimension for examples. It is easy to see that the existence of a non–linear similarity V_{1} ∼_{t} V_{2} implies dim V_{i} ≧ 5. Cappell, Shaneson, Steinberger and West [7], proved that 6-dimensional similarities exist for G = C(2^{r}), r ≧ 4, and asserted that 5– dimensional similarities do not exist for any finite group. On the other hand, the reference cited [5] has not yet appeared. In Theorem A we give a reduction of this question to an explicit number theory criterion. Using this criterion, we show in Corollary 19.2 that cyclic 2–groups do not have any 5–dimensional non– linear similarities. We have not been able to supply either a complete proof or a counterexample in the general case.

Table of Contents

§1. Introduction
§2. Statement of results
§3. A criterion for nonlinear similarity
§4. A splitting of R_{Top}(G)
§5. A rational computation
§6. Excision in bounded surgery Theory
§7. Bounded R_{−} transfers
§8. Calculations in bounded K-theory
§9. The double coboundary
§10. Some basic facts in K- and L-theory
§11. The computation of L_{1}^{p}(ZG, w)
§12. The proof of Theorem A
§13. The proof of Theorem B
§14. Calculations in bounded L–theory
§15. The proof of Theorem C
§16. Iterated R_{−} transfers
§17. The proof of Corollary 2.4.
§18. The normal invariant.
§19. Cyclic 2–groups.
§20. The proof of Theorem D

2. Statement of Results

We first introduce some notation, and then give the main results. Let G = C(4q), where q > 1, and let H = C(2q) denote the subgroup of index 2 in G. The maximal odd order subgroup of G is denoted G_{odd}. We fix a generator G = ⟨t⟩ and a primitive 4q^{th}-root of unity ζ = exp 2πi/4q. The group G has both a trivial 1-dimensional real representation, denoted R_{+}, and a non-trivial 1-dimensional real representation, denoted R_{−}.

A free G–representation is a sum of faithful 1-dimensional complex representations. Let t^{a}, a ∈ Z, denote the complex numbers C with action t · z = ζ^{a}z for all z ∈ C. This representation is free if and only if (a, 4q) = 1, and the coefficient a is well–defined only modulo 4q. Since t^{a} ≅ t^{−a} as real G–representations, we can always choose the weights a ≡ 1 (mod 4). This will be assumed unless otherwise mentioned.

Now suppose that V_{1} = t^{a_{1}} + ⋯ + t^{a_{k}} is a free G–representation. The Reidemeister torsion invariant of V_{1} is defined as

Δ(V_{1}) = ∏_{i=1}^{k}(t^{a_{i}} − 1) ∈ Z[t]/{±t^{m}} .

If V_{2} = t^{b_{1}} + ⋯ + t^{b_{k}} is another free representation, such that S(V_{1}) and S(V_{2}) are G-homotopy equivalent, then

Δ(V_{1})/Δ(V_{2}) = \frac{∏(t^{a_{i}} − 1)}{∏(t^{b_{i}} − 1)} ∈ Wh(ZG)

is the Whitehead torsion of any G-homotopy equivalence f: S(V_{2}) → S(V_{1}) [33]. When there exists a G-homotopy equivalence f: S(V_{2}) → S(V_{1}) which is normally cobordant to the identity map on S(V_{1}), we say that S(V_{1}) and S(V_{2}) are normally cobordant. More generally, we say that S(V_{1}) and S(V_{2}) are s–normally cobordant if S(V_{1} ⊕ U) and S(V_{2} ⊕ U) are normally cobordant for all free G–representations U. This is a necessary condition for non-linear similarity, which can be decided by explicit congruences in the weights.

This quantity, Δ(V_{1})/Δ(V_{2}) is the basic invariant determining non-linear similarity. To state concrete results we need to evaluate this invariant modulo suitable indeterminacy.

The involution t ↦ t^{−1} induces the identity on Wh(ZG), so we get an element

{Δ(V_{1})/Δ(V_{2})} ∈ H^{0}(Wh(ZG))

where we use H^{i}(A) to denote the Tate cohomology H^{i}(Z/2; A) of Z/2 with coefficients in A.

Let Wh(ZG^{−}) denote the Whitehead group Wh(ZG) together with the involution induced by t ↦ −t^{−1}. Then for τ(t) = \frac{∏(t^{a_{i}} −1)}{∏(t^{b_{i}} −1)}, we compute

τ(t)τ(−t) = \frac{∏(t^{a_{i}} − 1) ∏((−t)^{a_{i}} − 1)}{∏(t^{b_{i}} − 1) ∏((−t)^{b_{i}} − 1)} = ∏ \frac{(t^{2})^{a_{i}} − 1}{((t^{2})^{b_{i}} − 1)}

which is clearly induced from Wh(ZH). Hence we also get a well defined element

{Δ(V_{1})/Δ(V_{2})} ∈ H^{1}(Wh(ZG^{−})/ Wh(ZH)) .

Consider the exact sequence of Z/2 modules:

(2.1)  K_{1}(ZH) → K_{1}(ZG) → K_{1}(ZH →ZG) → K̃_{0}(ZH) → K̃_{0}(ZG)

and let Wh(ZH → ZG) = K_{1}(ZH → ZG)/{±G} . We then have a short exact sequence

0 → Wh(ZG)/ Wh(ZH) → Wh(ZH →ZG) → k → 0

where k = ker(K̃_{0}(ZH) → K̃_{0}(ZG)). Such an exact sequence of Z/2-modules induces a long exact sequence in Tate cohomology. In particular, we have a coboundary map

δ: H^{0}(k) → H^{1}(Wh(ZG^{−})/ Wh(ZH)) .

Theorem A. Let V_{1} = t^{a_{1}} + ⋯ + t^{a_{k}} and V_{2} = t^{b_{1}} + ⋯ + t^{b_{k}} be free G– representations, with a_{i} ≡ b_{i} ≡ 1 (mod4). There exists a topological similarity V_{1} ⊕ R_{−} ∼_{t} V_{2} ⊕ R_{−} if and only if

(i) ∏ a_{i} ≡ ∏ b_{i} (mod 4q), (ii) Res_{H} V_{1} ≅ Res_{H} V_{2}, and (iii) the element {Δ(V_{1})/Δ(V_{2})} ∈ H^{1}(Wh(ZG^{−})/ Wh(ZH)) is in the image of

the coboundary δ: H^{0}(k) → H^{1}(Wh(ZG^{−})/ Wh(ZH)).

Remark 2.2. Theorem A should be compared with [3,Cor.1], where more explicit conditions are given for similarities of this kind under the assumption that q is odd, or a 2-power, or 4q is a “tempered” number. See also Theorem 15.2 for a more general result concerning similarities without R_{+} summands. The case dim V_{1} = dim V_{2} = 4 gives a reduction to number theory for the existence of 5dimensional similarities (see Remark 12.2). □

Our next result uses a more elaborate setting for the invariant. Let

Φ = ( \begin{matrix}ZH & → & Ẑ_{2}H \\ ↓ & & ↓ \\ ZG & → & Ẑ_{2}G \end{matrix} )

and consider the exact sequence

(2.3) 0 → K_{1}(ZH →ZG) → K_{1}(Ẑ_{2}H →Ẑ_{2}G) → K_{1}(Φ) → K̃_{0}(ZH →ZG) → 0 .

Again we can define the Whitehead group versions by dividing out trivial units {±G}, and get a double coboundary

δ^{2}: H^{1}(K̃_{0}(ZH →ZG^{−})) → H^{1}(Wh(ZH →ZG^{−})) .

There is a natural map H^{1}(Wh(ZG^{−})/ Wh(ZH)) → H^{1}(Wh(ZH →ZG^{−})).

Theorem B. Let V_{1} = t^{a_{1}} + ⋯ + t^{a_{k}} and V_{2} = t^{b_{1}} + ⋯ + t^{b_{k}} be free G– representations. There exists a topological similarity V_{1}⊕R_{−}⊕R_{+} ∼_{t} V_{2}⊕R_{−}⊕R_{+} if and only if

(i) ∏ a_{i} ≡ ∏ b_{i} (mod 4q),

(ii) Res_{H} V_{1} ≅ Res_{H} V_{2}, and

(iii) the element {Δ(V_{1})/Δ(V_{2})} is in the image of the double coboundary

δ^{2}: H^{1}(K̃_{0}(ZH →ZG^{−})) → H^{1}(Wh(ZH →ZG^{−})) .

This result can be applied to 6-dimensional similarities.

Corollary 2.4. Let G = C(4q), with q odd, and suppose that the fields Q(ζ_{d}) have odd class number for all d | 4q. Then G has no 6-dimensional non-linear similarities.

Remark 2.5. This corrects [7,Thm.1(i)], and invalidates the computations of R_{Top}(G) given in [7,Thm.2]. □

Our final computation of the bounded transfer is suitable for determining stable non–linear similarities inductively. To state the conditions, we must again generalize the indeterminacy for the Reidemeister torsion invariant to include bounded K– groups (see Section 6). Since K_{i}(ZH →ZG) = K_{i}(C_{R_{−},G}(Z)), we can consider the analogous double coboundary

δ^{2}: H^{1}(K̃_{0}(C_{W ×R_{−},G}(Z))) → H^{1}(Wh(C_{W ×R_{−},G}(Z)))

and note that there is a map Wh(C_{R_{−},G}(Z) → Wh(C_{W ×R_{−},G}(Z)) induced by the inclusion on the control spaces, so our invariant {Δ(V_{1})/Δ(V_{2})} maps forward into this new domain.

Theorem C. Let V_{1} = t^{a_{1}} + ⋯ + t^{a_{k}} and V_{2} = t^{b_{1}} + ⋯ + t^{b_{k}} be free G– representations. Let W be a complex G–representation with no R_{+} summands. Then there exists a topological similarity V_{1} ⊕W ⊕R_{−} ⊕R_{+} ∼_{t} V_{2} ⊕W ⊕R_{−} ⊕R_{+} if and only if

(i) S(V_{1}) is s–normally cobordant to S(V_{2}), (ii) Res_{H}(V_{1} ⊕ W) ⊕ R_{+} ∼_{t} Res_{H}(V_{2} ⊕ W) ⊕ R_{+}, and (iii) the element {Δ(V_{1})/Δ(V_{2})} is in the image of the double coboundary

δ^{2}: H^{1}(K̃_{0}(C_{W_{max}×R_{−},G}(Z))) → H^{1}(Wh(C_{W_{max}×R_{−},G}(Z))) ,

where 0 ⊆ W_{max} ⊆ W is a 2–dimensional complex subrepresentation with maximal isotropy group among the isotropy groups of W with 2-power index.

Remark 2.6. Note that W_{max} = 0 in condition (iii) if W has no isotropy subgroups of 2-power index. Theorem C suffices to handle stable topological similarities, but leaves out cases where W has an odd number of R_{−} summands (see Section 16). Simpler conditions can be given when G = C(2^{r}) (see Section 19).

The double coboundary in (iii) can also be expressed in more “classical” terms by using the short exact sequence

(2.7) 0 → Wh(C_{R_{−},G}(Z)) → Wh(C_{W_{max}×R_{−},G}(Z)) → K_{1}(C_{W_{max}×R_{−},G}^{>R_{−}}(Z)) → 0

derived in Corollary 8.9. We have K_{1}(C_{W_{max}×R_{−},G}^{>R_{−}}(Z)) = K_{−1}(ZK), where K is the isotropy group of W_{max}, and Wh(C_{R_{−},G}(Z)) = Wh(ZH →ZG). The indeterminacy in Theorem C is then generated by the double coboundary

δ^{2}: H^{1}(K̃_{0}(ZH →ZG^{−})) → H^{1}(Wh(ZH →ZG^{−}))

used in Theorem B and the coboundary

δ: H^{0}(K_{−1}(ZK)) → H^{1}(Wh(ZH →ZG^{−}))

from the Tate cohomology sequence of (2.7). □

Finally, we will apply these results to R_{Top}(G). In Section 4, we will define a filtration

(2.8)  R_{t}(G) ⊆ R_{n}(G) ⊆ R_{h}(G) ⊆ R(G)

on the real representation ring R(G), inducing a filtration on

R_{Top}(G) = R(G)/R_{t}(G) .

Here R_{h}(G) consists of those virtual elements with no homotopy obstruction to similarity, and R_{n}(G) the virtual elements with no normal invariant obstruction to similarity (see Section 4 for more precise definitions). Note that R(G) has the nice basis {t^{i}, δ, ϵ | 1 ≦ i ≦ 2q − 1}, where δ = [R_{−}] and ϵ = [R_{+}].

Let R^{free}(G) = {t^{a} | (a, 4q) = 1} ⊂ R(G) be the subgroup generated by the free representations. To complete the definition, we let R^{free}(C(2)) = {R_{−}} and R^{free}(e) = {R_{+}}. Then

R(G) = ⨁_{K⊆G} R^{free}(G/K)

and this direct sum splitting can be intersected with the filtration above to define Rh^{free}(G), Rn^{free}(G) and Rt^{free}(G). In addition, we can divide out Rt^{free}(G) and obtain subgroups Rh,Top^{free}(G) and Rn,Top^{free}(G) of RTop^{free}(G) = R^{free}(G)/Rt^{free}(G). By induction on the order of G, we see that it suffices to study the summand RTop^{free}(G).

Let R̃^{free}(G) = ker(Res: R^{free}(G) → R^{free}(G_{odd})), and then project into R_{Top}(G) to define

R̃Top^{free}(G) = R̃^{free}(G)/Rt^{free}(G) .

In Section 5 we prove that R̃Top^{free}(G) is precisely the torsion subgroup of RTop^{free}(G), and the structure of this subgroup is determined in Section 20. Here is a sample computation.

Theorem D. Let G = C(4q), with q odd, and suppose that the fields Q(ζ_{d}) have odd class number for all d | 4q. Then R̃Top^{free}(G) = Z/4 generated by (t − t^{1+2q}).

For any cyclic group G, both R^{free}(G)/Rh^{free}(G) and Rh^{free}(G)/Rn^{free}(G) are torsion groups which can be explicitly determined by congruences in the weights [3] (but see [40] for a correction to [6,Condition A]). In Section 20 we show that the remaining subquotient R̃n,Top^{free}(G) = R̃n^{free}(G)/Rt^{free}(G) always has exponent two.

3. A criterion for non–linear similarity

Our approach to the non-linear similarity problem is through bounded surgery theory (see [18], [19]). First an elementary observation about topological equivalences for cyclic groups.

Lemma 3.1. If V_{1} ⊕ W ∼_{t} V_{2} ⊕ W, where V_{1}, V_{2} are free G–representations, and W and W′ have no free summands, then there is a G-homeomorphism h: V_{1} ⊕W → V_{2} ⊕ W such that

h| ⋃_{1≠H≤G} W^{H}

is the identity.

Proof. Let h be the homeomorphism. We will successively change h, stratum by stratum. For every subgroup K of G, consider the homeomorphism of K-fixed sets

f^{K}: W^{K} → W^{′K}.

This is a homeomorphism of G/K, hence of G–representations. As G–representations we can split

V_{2} ⊕ W′ = U ⊕ W^{′K} ∼_{t} U ⊕ W^{K} = V_{2} ⊕ W′′

where the similarity uses the product of the identity and (f^{K})^{−1}. Rename W′′ and repeat this successively for all subgroups. We end up with W = W′ and a homeomorphism as claimed. □

One consequence is

Lemma 3.2. If V_{1} ⊕ W ∼_{t} V_{2} ⊕ W then there exists a G-homotopy equivalence S(V_{2}) → S(V_{1}).

Proof. If we 1-point compactify h we obtain a G-homeomorphism

h^{+}: S(V_{1} ⊕ W ⊕ R) → S(V_{2} ⊕ W ⊕ R).

After an isotopy, the image of the free G-sphere S(V_{1}) may be assumed to lie in the complement S(V_{2} ⊕ W ⊕ R) − S(W ⊕ R) of S(W ⊕ R) which is G-homotopy equivalent to S(V_{2}). □

Any homotopy equivalence f: S(V_{2})/G → S(V_{1})/G defines an element [f] in the structure set S^{h}(S(V_{1})/G). We may assume that dim V_{i} ≧ 4. This element must be non-trivial: otherwise S(V_{2})/G would be topologically h-cobordant to S(V_{1})/G, and Stallings infinite repetition of h-cobordisms trick would produce a homeomorphism V_{1} → V_{2} contradicting [1], since V_{1} and V_{2} are free representations. More precisely, we use Wall’s extension of the Atiyah–Singer equivariant index formula to the topological locally linear case [38]. If dim V_{i} = 4, we can cross with CP^{2} to avoid low–dimensional difficulties. Crossing with W and parameterising by projection on W defines a map from the classical surgery sequence to the bounded surgery exact sequence

(3.3) \xymatrix{L_{n}^{h}(ZG) \ar[r] \ar[d] & S^{h}(S(V_{1})/G) \ar[r] \ar[d] & [S(V_{1})/G, F/Top] \ar[d] \\ L_{n+k}^{h}(C_{W,G}(Z)) \ar[r] & S_{b}^{h} ( \begin{matrix}S(V_{1}) × W/G \\ ↓ \\ W/G \end{matrix} ) \ar[r] & [S(V_{1}) ×_{G} W, F/Top]}

The L–groups in the upper row are the ordinary surgery obstruction groups for oriented manifolds and surgery up to homotopy equivalence. In the lower row, we have bounded L–groups (see Section 6) corresponding to an orthogonal action ρ_{W}: G → O(W), with orientation character given by det(ρ_{W}). Our main criterion for non–linear similarities is:

Theorem 3.4. Let V_{1} and V_{2} be free G–representations with dim V_{i} ≧ 2, and suppose that f: S(V_{2}) → S(V_{1}) is a G–homotopy equivalence. Then, there is a topological equivalence V_{1} ⊕ W ∼_{t} V_{2} ⊕ W if and only if the element [f] ∈ S^{h}(S(V_{1})/G) is in the kernel of the bounded transfer map

trf_{W}: S^{h}(S(V_{1})/G) → S_{b}^{h}( \begin{matrix}S(V_{1}) ×_{G} W \\ ↓ \\ W/G \end{matrix} ) .

Proof. For necessity, we refer the reader to [19] where this is proved using a version of equivariant engulfing. For sufficiency, we notice that crossing with R gives an isomorphism of the bounded surgery exact sequences parameterized by W to simple bounded surgery exact sequence parameterized by W × R. By the bounded scobordism theorem, this means that the vanishing of the bounded transfer implies that

\xymatrix{S(V_{2}) × W × R \ar[r]^{f×1} & S(V_{1}) × W × R \ar[d] \\  & W × R}

is within a bounded distance of an equivariant homeomorphism h, where distances are measured in W × R. We can obviously complete f × 1 to the map

f ∗ 1: S(V_{2}) ∗ S(W × R) → S(V_{1}) ∗ S(W × R)

and since bounded in W × R means small near the subset

S(W × R) ⊂ S(V_{i}) ∗ S(W × R) = S(V_{i} ⊕ W ⊕ R),

we can complete h by the identity to get a homeomorphism

S(V_{2} ⊕ W ⊕ R) → S(V_{1} ⊕ W ⊕ R)

and taking a point out we have a homeomorphisms V_{2} × W → V_{2} × W □

By comparing the ordinary and bounded surgery exact sequences (3.3), and noting that the bounded transfer induces the identity on the normal invariant term, we see that a necessary condition for the existence of any stable similarity f: V_{2} ≈_{t} V_{1} is that f: S(V_{2}) → S(V_{1}) has s–normal invariant zero. Assuming this, under the natural map

L_{n}^{h}(ZG) → S^{h}(S(V_{1})/G),

where n = dim V_{1}, the element [f] is the image of σ(f) ∈ L_{n}^{h}(ZG), obtained as the surgery obstruction (relative to the boundary) of a normal cobordism from f to the identity. The element σ(f) is well-defined in L̃_{n}^{h}(ZG) = Coker(L_{n}^{h}(Z) → L_{n}^{h}(ZG)).

Theorem 3.5. Let V_{1} and V_{2} be free G–representations with dim V_{i} ≧ 2, and suppose that f: S(V_{2}) → S(V_{1}) is a G–homotopy equivalence which is G–normally cobordant to the identity. Then, there is a topological equivalence V_{1} ⊕W ∼_{t} V_{2} ⊕W if and only if trf_{W} (σ(f)) = 0, where trf_{W}: L_{n}^{h}(ZG) → L_{n}^{h}(C_{W,G}(Z)) is the bounded transfer.

The rest of the paper is about the computation of these bounded transfer homomorphisms in L–theory. We will need the following result (proved for K_{0} in [19,6.3]).

Theorem 3.6. Let W be a G–representation with W^{G} = 0. For all i ∈ Z, the image of the bounded transfer trf_{W}: K_{i}(ZG) → K_{i}(C_{W,G}(Z)) is contained in the image of the cone point inclusion c_{∗}: K_{i}(ZG) = K_{i}(C_{pt,G}(Z)) → K_{i}(C_{W,G}(Z)).

Proof. Let G be a finite group and V a representation. Crossing with V defines a transfer map in K–theory K_{i}(RG) → K_{i}(C_{V,G}(R)) for i ≤ 1, where R is any ring with unit. This map is actually defined for all i, and turns out to be equal to the map K_{i}(C_{0,G}(R)) → K_{i}(C_{V,G}(R)) induced by the inclusion 0 ⊂ V. To see this we need to choose models for K–theory. For RG we choose the category of finitely generated free RG modules, but we think of it as a category with cofibrations and weak equivalences with weak equivalences isomorphisms and cofibrations split inclusions. For C_{V,G}(R) we use the category of finite length chain complexes, with weak equivalences chain homotopy equivalences and cofibrations sequences that are split short exact at each level. The K–theory of this category is the same as the K-theory of C_{V,G}(R). For an argument working in this generality see [8]. Tensoring with the chain complex of (V, G) induces a map of categories with cofibrations and weak equivalences hence a map on K–theory. It is elementary to see this agrees with the geometric definition in low dimensions, since the identification of the K– theory of chain complexes of an additive category with the K–theory of the additive category is an Euler characteristic (see e. g. [8]). By abuse of notation we denote the category of finite chain complexes in C_{V,G}(R) simply by C_{V,G}(R). We need to study various related categories. First there is C_{V,G}^{iso}(R) where we have replaced the weak equivalences by isomorphisms. Obviously the transfer map, tensoring with the chains of (V, G) factors through this category. Also the transfer factors through the category D_{V,G}^{iso}(R) with the same objects, and isomorphisms as weak equivalences but the control condition is 0-control instead of bounded control. The category D_{V,G}^{iso}(R) is the product of the full subcategories on objects with support at 0 and the full subcategory on objects with support on V − 0, D_{0,G}^{iso}(R) × D_{V −0,G}^{iso}(R), and the transfer factors through chain complexes concentrated in degree 0 in D_{0,G}^{iso}(R) crossed with chain complexes in the other factor. But the subcategory of chain complexes concentrated in degree zero of D_{0,G}^{iso}(R) is precisely the same as C_{0,G}(R) and the map to C_{V,G}(R) is induced by inclusion. So to finish the proof we have to show that the other factor D_{V −0,G}^{iso}(R) maps to zero. For this we construct an intermediate category E_{V −0,G}^{iso}(R) with the same objects, but where the morphisms are bounded radially and 0-controlled otherwise (i. e. a nontrivial map between objects at different points is only allowed if the points are on the same radial line, and there is a bound on the distance independent of the points). This category has trivial K-theory since we can make a radial Eilenberg swindle toward infinity. Since the other factor D_{V −0,G}^{iso}(R) maps through this category, we find that the transfer maps through the corner inclusion as claimed.

Remark 3.7. It is an easy consequence of the filtering arguments based on [18,Thm. 3.12] that the bounded L–groups are finitely generated abelian groups with 2– primary torsion subgroups. We will therefore localize all the L–groups by tensoring with Z_{(2)} (without changing the notation), and this loses no information for computing bounded transfers. One concrete advantage of working with the 2–local L–groups is that we can use the idempotent decomposition [15,§6] and the direct sum splitting L_{n}^{h}(C_{W,G}(Z)) = ⊕_{d|q}L_{n}^{h}(C_{W,G}(Z))(d). Since the “top component” L_{n}^{h}(C_{W,G}(Z))(q) is just the kernel of the restriction map to all odd index subgroups of G, the use of components is well-adapted to inductive calculations. □

A first application of these techniques was given in [19,5.1].

Theorem 3.8. For any G–representation W, let W = W_{1} ⊕ W_{2} where W_{1} is the direct sum of the irreducible summands of W with isotropy subgroups of 2–power index. If W^{G} = 0, then

(i) the inclusion L_{n}^{h}(C_{W_{1},G}(Z))(q) → L_{n}^{h}(C_{W,G}(Z))(q) is an isomorphism on the top component,

(ii) the bounded transfer

trf_{W_{2}}: L_{n}^{h}(C_{W_{1},G}(Z))(q) → L_{n}^{h}(C_{W,G}(Z))(q)

is an injection on the top component, and (iii) ker(trf_{W}) = ker(trf_{W_{1}}) ⊆ L_{n}^{h}(ZG)(q).

Proof. In [19] we localized at an odd prime p ∤ |G| in order to use the Burnside idempotents for all cyclic subgroups of G. The same proof works for the L–groups localized at 2, to show that trf_{W_{2}} is injective on the top component. □

Lemma 3.9. For any choice of normal cobordism between f and the identity, the surgery obstruction σ(f) is a nonzero element of infinite order in L̃_{n}^{h}(ZG).

Proof. See [19,4.5] □

The following result (combined with Theorem 3.5) shows that there are no nonlinear similarities between semi-free G–representations, since L_{n+1}^{h}(C_{R,G}(Z)) = L_{n}^{p}(ZG) and the natural map L_{n}^{h}(ZG) → L_{n}^{p}(ZG) may be identified with the bounded transfer trf_{R}: L_{n}^{h}(ZG) → L_{n+1}^{h}(C_{R,G}(Z)) [31,§15].

Corollary 3.10. Under the natural map L_{n}^{h}(ZG) → L_{n}^{p}(ZG), the image of σ(f) is nonzero.

Proof. The kernel of the map L_{n}^{h}(ZG) → L_{n}^{p}(ZG) is the image of H^{n}(K̃_{0}(ZG)) which is a torsion group. □

4. A splitting of R_{Top}(G)

In this section we point out an elementary splitting of R_{Top}(G), and some useful filtrations.

For G any finite group, we denote by R(G) the real representation ring of G. Elements in R(G) can be given as formal differences (V_{1} −V_{2}) of G–representations, and (V_{1} −V_{2}) ∼ 0 if and only if there exists a representation W such that V_{1} ⊕W ≅ V_{2} ⊕ W.

Notice that for K any normal subgroup of G, taking fixed sets gives a retraction of the inflation map

Inf_{K}: R(G/K) → R(G)

defined by pulling back a G/K representation using the composition with the quotient map G → G/K. More explicitly,

Fix_{K}: R(G) → R(G/K)

is defined by Fix_{K}(V_{1} − V_{2}) = (V_{1}^{K} − V_{2}^{K}) for each normal subgroup K ◁ G. Then Fix_{K} ∘ Inf_{K} = id: R(G/K) → R(G/K).

Definition 4.1. A G–representation V is free if V^{K} = {0} for all non–trivial normal subgroups 1 ≠ K ◁ G.

This is the same as the usual definition (no non–identity element of G fixes any non–zero vector) for cyclic groups. We let

(4.2)  R^{free}(G) = {(V_{1} − V_{2}) | V_{1}, V_{2} free representations}

denote the subgroup of R(G) such that there is a representative of the stable equivalence class (V_{1} − V_{2}) so that V_{1}, V_{2} are free.

Proposition 4.3. There is a direct sum splitting

R(G) = ⨁_{K◁G} R^{free}(G/K)

indexed by the normal subgroups K in G.

Inside R(G) have the subgroup of stably topologically similar representations

(4.4)  R_{t}(G) = {(V_{1} − V_{2}) | V_{1} ⊕ W ∼_{t} V_{2} ⊕ W for some W }

and the quotient group is R_{Top}(G) by definition. Since this subgroup is preserved by inflation and taking fixed sets, we obtain Corollary 4.5. There is a direct sum decomposition

R_{Top}(G) = ⨁_{K◁G} R^{free}_{Top}(G/K)

where the summand are the quotients R^{free}(G/K)/Rt^{free}(G/K).

We will also need a certain filtration of R(G). First we define

(4.6)  R_{h}(G) = {(V_{1} − V_{2}) | S(V_{1} ⊕ W) ≃_{G} S(V_{2} ⊕ W) for some W }

where ≃_{G} denotes G–homotopy equivalence. This is a subgroup of R(G), in fact a sub-Mackey functor since it has induction and restriction for subgroups of G. If there exists a G–homotopy equivalence f: S(V_{1} ⊕ W) → S(V_{2} ⊕ W) such that

S(V_{1} ⊕ W ⊕ U) = S(V_{1} ⊕ W)∗S(U)⟶^{f∗1}S(V_{1} ⊕ W)∗S(U) = S(V_{1} ⊕ W ⊕ U)

which is G–normally cobordant to the identity for all G–representations U, then we write S(V_{1} ⊕ W) ≏_{G} S(V_{2} ⊕ W). Define

(4.7)  R_{n}(G) = {(V_{1} − V_{2}) | S(V_{1} ⊕ W) ≏_{G} S(V_{2} ⊕ W) for some W }

and note that R_{n}(G) is a subgroup of R(G). Indeed, if (V_{1} − V_{2}) and (V′_{1} − V′_{2}) are in R_{n}(G), there exist G-homotopy equivalences f: S(V_{1} ⊕ W) → S(V_{2} ⊕ W) and f′: S(V′_{1} ⊕W′) → S(V′_{2} ⊕W′) with f∗1 and f′∗1 normally cobordant to the identity under any stabilization. But f∗f′ ≃_{G} (1∗f′) ∘ (f∗1), so we just glue together the normal cobordisms for f∗1 (after stabilizing by U = V′_{1} ⊕ W′) and for 1∗f′ (after stabilizing by U = V_{2} ⊕ W) along the common boundary id: S(V_{2} ⊕ W ⊕ V′_{1} ⊕ W′).

Since R_{t}(G) ⊆ R_{n}(G), we have defined a filtration

(4.8)  R_{t}(G) ⊆ R_{n}(G) ⊆ R_{h}(G) ⊆ R(G)

of R(G), natural with respect to induction and restriction of representations. Now we intersect this filtration with the splitting described above. Define a subgroup of R(G) and subgroups for each decoration (?) = h, n, t by

(4.9)  R(?)^{free}(G) = {(V_{1} − V_{2}) | V_{1}, V_{2} free representations}

or, in other words, we assume that there is a representative of the stable equivalence class (V_{1} − V_{2}) so that V_{1}, V_{2} are free.

Lemma 4.10. The subfiltration R(?)^{free}(G) is well–defined.

Proof. Any V has a unique direct sum splitting V = V_{1}⊕W with V_{1} free of maximal dimension (V_{1} = 0 is possible). Any G-homotopy equivalence ϕ: S(V_{1}⊕W → S(V_{2}⊕ W) is stably G-homotopic to one of the form f∗1, where f: S(V_{2}) → S(V_{1}) is a Ghomotopy equivalence on the free parts [11]. Since homotopy equivalences of free G–spheres are detected by the first k–invariant (which is multiplicative under joins [38,14E]), we note that S(V_{1} ⊕U) ≃_{G} S(V_{2} ⊕U) if and only if S(V_{1}) ≃_{G} S(V_{2}) when V_{1}, V_{2} and U are free representations. It follows that Rh^{free}G is just the subgroup of virtual elements (V_{1} − V_{2}) with V_{1}, V_{2} free, and S(V_{1}) ≃_{G} S(V_{2}). Similarly, (V_{1} − V_{2}) ∈ Rn^{free}G if and only if there exists a free G-homotopy equivalence f: S(V_{2}) → S(V_{1}) such that f∗1: S(V_{1} ⊕ U) → S(V_{2} ⊕ U) is freely G-normally cobordant to the identity for all free representations U. □

Using the groups R(?)^{free}(G/K) we can refine our previous splitting.

Lemma 4.11. For R(G) and (?) = h, n, t there is a direct sum splitting

R_{(?)}(G) = ⨁_{K◁G} R(?)^{free}(G/K)

so that the filtration (4.8) is the direct sum of the filtrations on the free parts.

Proof. Just checking through the definitions. □

We conclude this list of definitions by pointing out an even finer direct sum splitting of R(G) for G cyclic, after localizing at 2, using Mackey functor structure. The components are indexed by the family O(G) of subgroups C ⊂ G of odd order [15,§6], [25,11b]. Let q = q(G) denote the odd part of the order of G, then for (?) = h, n, t we have

(4.12)  R_{(?)}(G) ⊗ Z_{(2)} = ⨁_{d|q} R_{(?)}(G)(d)

We call this splitting the Burnside splitting (or the 2–local Burnside splitting if we need to distinguish it from the corresponding splittings at other primes). It is induced by the action of idempotents e_{C} in the 2–local Burnside ring.

Proposition 4.13. Let G be a cyclic group. The Burnside splitting of R(G) induces a splitting

R(?)^{free}(G/K) ⊗ Z_{(2)} = ⨁_{d|q(G/K)} R(?)^{free}(G/K)(d)

for (?) = h, n, t compatible with the splitting of (4.8) into free summands.

Proof. We must compare the operation of the idempotents e_{C} with the splitting maps Inf_{K}: R(G/K) → R(G) and Fix_{K}: R(G) → R(G/K). The cyclic subgroups of G/K are just the cyclic subgroups of G containing K. Therefore the maps e_{C} commute with Inf_{K} and Fix_{K} whenever C is a cyclic subgroup of G containing K. □

5. A rational computation

In this section we use Theorem 3.5 and the splitting of the last section to describe the torsion subgroup of RTop^{free}(G). We also give a new proof of Cappell and Shaneson’s result computing R_{Top}(G) ⊗ Q for all finite groups G.

First we consider cyclic groups. Let G = C(2^{r}q) be a cyclic group, where q ≧ 1 is odd. Notice that the Odd Order Theorem [23], [24] gives R(G) = R_{Top}(G) if r ≦ 1. Recall that

R̃Top^{free}(G) = R̃^{free}(G)/Rt^{free}(G)

where R̃^{free}(G) = ker{Res: R^{free}(G) → R^{free}(G_{odd})}. Notice that R̃^{free}(C(2q)) = R̃^{free}(C(q)) = 0, for q ≧ 1 odd.

Theorem 5.1. For G cyclic, the kernel and cokernel of

Res: RTop^{free}(G) → RTop^{free}(G_{odd}) = R^{free}(G_{odd})

are 2–primary torsion groups.

Corollary 5.2. Let G = C(2^{r}q), q odd, be a finite cyclic group.

(i) The torsion subgroup of RTop^{free}(G) is R̃Top^{free}(G).

(ii) The rank of RTop^{free}(G) is φ(q)/2 (resp. 1) if q > 1 (resp. q = 1).

(iii) We have the formula

rank(R_{Top}(G)) = \{ \begin{matrix}(r + 1)\{∑_{d|q}(φ(d)/2 + 1)\}, & for q > 1 \\ (r + 1) & for q = 1. \end{matrix}

where φ(d) is the Euler function.

Proof. The first part follows directly from Theorem 5.1. For parts (ii) and (iii) count the free representations of G_{odd}. □

Now let F be the composite of all subfields of R of the form Q(ζ + ζ^{−1}) where ζ ∈ C is an odd root of unity.

Lemma 5.3. For G cyclic, the composition of the natural map R_{F} (G) → R(G) and Res: R(G) → R(G_{odd}) induces a p–local isomorphism RF^{free}(G) → R^{free}(G_{odd}), for any odd prime p.

Proof. According to a result of Brauer [35,Thm.24] R_{F} (G_{odd}) = R(G_{odd}), and any representation of G can be realized over the field Q(ζ_{|G|}). In addition, the restriction map

Res: R_{F} (G) → R_{F} (G_{odd})

is a p–local surjection (since Res_{G_{odd}} ∘ Ind_{G_{odd}} is just multiplication by 2^{r}). But the rank of R_{F} (G) given in [35,12.4] equals the rank of R(G_{odd}), so we are done. □

Corollary 5.4. For G cyclic and any odd prime p, the natural map R_{F} (G) → R_{Top}(G) induces a p–local isomorphism.

Proof. This follows from the Lemma 5.3 and Theorem 5.1. □

Proof of Theorem 5.1. Since a F –representation is free if and only if it is the sum of Galois conjugates of free G–representations, we can decompose R_{F} (G) as in Section 4, and conclude that RF^{free}(G_{odd}) = R^{free}(G_{odd}). It remains to show that R̃Top^{free}(G) is a torsion group with 2–primary exponent. For this we use the filtration of §4, and claim that when intersected with R̃Top^{free}(G) each successive quotient is 2–primary torsion.

For R̃^{free}(G)/R̃h^{free}(G) this is easy since the k–invariant gives a homomorphism (via joins of free G–spheres) to (Z/2^{r})^{×} and this is a 2–group. The next quotient is also 2–primary torsion, by results of [40]: a sufficiently large 2–power join of a G–homotopy equivalence between two free G–spheres, which are linearly equivalent over G_{odd}, becomes s–normally cobordant to the identity. The point is that the normal invariant is detected by a finite number of 2–power congruences conditions among the Hirzebruch L–classes of the tangent bundles of the lens spaces, and this can be satisfied after sufficiently many joins.

Finally, the last quotient R̃n^{free}(G)/R̃t^{free}(G) is shown to be 2–primary torsion in the next proposition. □

Proposition 5.5. Let G = C(2^{r}q), q odd, and assume

σ ∈ ker(Res: L_{2k}^{h}(ZG) → L_{2k}^{h}(ZG_{odd}).

Then there exists a complex representation W with W^{G} = 0 such that trf_{W} (2^{r}σ) = 0.

Proof. We will take W = R_{−} ⊕R_{−} ⊕W_{0}, where W_{0} is the sum of all the irreducible 2–dimensional representations of G with isotropy of 2–power index. Note that the R_{−}-transfer is just the compact I_{−} transfer of one-sided codimension 1 surgery followed by adding rays to infinity, so whenever the I_{−}transfer is 0, the R_{−}-transfer will have to be 0. This will be discussed in more detail in Section 7. Step 1: If G has odd order, there is nothing to prove. Otherwise, let H ⊂ G be of index 2. If Res_{H}(σ) = 0, then trf_{R_{−}}(σ) ∈ Im(L_{1}^{h}(ZG^{−}) → L_{1}^{h}(ZH → ZG^{−}). But L_{1}^{h}(ZG^{−}) has exponent 2 [20], so trf_{R_{−}}(2σ) = 0. Then take W = R_{−} ⊕ R_{−}. Note that this case applies to G = C(2q), so we can always get started. Step 2: We may assume that r ≧ 2. If Res_{H}(σ) ≠ 0 note that

Res_{H}(2σ − Ind_{H} Res_{H}(σ)) = 2 Res_{H}(σ) − 2 Res_{H}(σ) = 0.

By induction Res_{H} W_{0} works for Res_{H}(σ): say

trf_{Res_{H} W_{0}}(2^{r−1} Res_{H}(σ)) = 0

and W_{0}^{H} = 0. Let dim W_{0} = m and consider the commutative diagram

\xymatrix{L_{2k}(ZG) \ar[r]^{Res} \ar[d]_{_{trf_{W_{0}}}} & L_{2k}(ZH) \ar[r]^{Ind} \ar[d]_{_{trf_{Res_{H} W_{0}}}} & L_{2k}(ZG) \ar[d]_{_{trf_{W_{0}}}} \\ L_{2k+m}(C_{W_{0},G}(Z)) \ar[r]^{Res} & L_{2k+m}(C_{Res W_{0},H}(Z)) \ar[r]^{Ind} & L_{2k+m}(C_{W_{0},G}(Z)).}

From this we get

2^{r−1} · trf_{W_{0}}(Ind_{H} Res_{H}(σ)) = 0 .

The first step implies that trf_{R_{−}^{2}}(2σ_{1}) = 0, where σ_{1} = 2σ − Ind_{H} Res_{H}(σ). Let W = R_{−} ⊕ R_{−} ⊕ W_{0}, so that we have W complex and W^{G} = 0. Note that trf_{W} = trf_{R_{−}^{2}} ∘ trf_{W_{0}} = trf_{W_{0}} ∘ trf_{R_{−}^{2}}. But

2^{r} · trf_{W} (σ) = 2^{r−1} trf_{W} (2σ − Ind_{H} Res_{H}(σ)) + 2^{r−1} trf_{W} (Ind_{H} Res_{H}(σ))

and both terms vanish (because r ≧ 2 and by the property of W_{0} respectively). □

A similar argument to that in Step 2 above gives:

Proposition 5.6. If Res_{H}(trf_{W} (x)) = 0 for x ∈ L_{0}^{h}(ZG), then 4 · trf_{W ×R_{−}}(x) = 0.

Proof. Since

4·trf_{W ×R_{−}}(σ) = 2 trf_{W} trf_{R_{−}}(2σ −Ind_{H} Res_{H}(σ))+2 trf_{R_{−}} trf_{W} (Ind_{H} Res_{H}(σ))

we conclude as above that both terms vanish. □

In [4], Cappell and Shaneson obtained the following result by a different argument.

Theorem 5.7. Let F be the composite of all subfields of C of the form Q(ζ + ζ^{−1}) where ζ ∈ C is an odd root of unity. Then the natural map R_{F} (G) → R(G) induces an isomorphism R_{F} (G) ⊗ Z_{(p)} ≅ R_{Top}(G) ⊗ Z_{(p)} for any odd prime p.

Proof. The result holds for cyclic groups G by Corollary 5.4, and we apply induction theory to handle general finite groups.

First we observe that the Mackey functors R(G) ⊗ Z_{(p)} and R_{F} (G) ⊗ Z_{(p)} are generated by induction from p–elementary subgroups (see [35,Thm.27] and note that Γ_{F} = {±1}). Therefore, by [25,11.2] they are p–elementary computable in the sense of Dress induction theory [12]. We may therefore assume that G is p– elementary.

Now if G is p–elementary, it is a product of a p–group and a cyclic group prime to p. Any irreducible complex representation of G is then induced from a linear representation on a subgroup [35,Thm.16].

It follows that R_{Top}(G) is generated by generalized induction (i.e. inflation followed by induction) from cyclic subquotients. Consider the following commutative diagram

[figure: R_{F} (G); Res; R(G); R_{Top}(G); ⨁_{C}R_{F} (C); Ind; ⨁_{C}R(C); ⨁_{C}R_{Top}(C)]

It follows from Corollary 5.4 that the top composite map R_{F} (G)⊗Z_{(p)} → R_{Top}(G))⊗ Z_{(p)} is surjective.

The sum of the (ordinary) restriction maps to cyclic subgroups induces a rational injection on R_{F} (G) (see [37,2.5,2.10]). Since R_{F} (G) is torsion–free, it follows again from Corollary 5.4 that the map R_{F} (G) ⊗ Z_{(p)} → R_{Top}(G)) ⊗ Z_{(p)} is injective. □

6. Excision in bounded surgery theory

A small additive category with involution A is a small additive category together with a contravariant endofunctor ∗ such that ∗^{2} = 1_{A}. Ranicki defines algebraic L–theory L_{∗}^{h}(A) for such categories and corresponding spectra 𝕃^{h}(A) with L_{∗}^{h}(A) = π_{∗}(𝕃^{h}(A)) [29]. The obstruction groups for bounded surgery are obtained this way for appropriately chosen additive categories. We shall also need a simple version of such groups. For this, the additive category must come equipped with a system of stable isomorphisms and a subgroup s ⊂ K_{1}(A), such that any composition resulting in an automorphism defines an element in s. The point here is that whenever two objects are stably isomorphic, there is a canonically chosen stable isomorphism, canonical up to automorphisms defining elements of a. In this situation Ranicki refines the definition of 𝕃^{h}(A) to give the simple L–theory spectrum 𝕃^{s}(A), by requiring appropriate isomorphisms to give elements of K_{1}(A) belonging to the subgroup s. More generally, We also get the 𝕃\overline{s}(A)-spectra for any involution invariant subgroup a with s ⊂ a ⊂ K_{1}(A), coinciding with 𝕃^{h}(A) when a = K_{1}(A).

Example 6.1. Let A be the category of free ZG-modules with a G-invariant Zbasis, and ZG-module morphisms. Two objects are stably isomorphic if and only if they have the same rank. The preferred isomorphisms are chosen to be the ones sending a Z-basis to a Z-basis, so automorphisms define elements of {±G} ⊂ K_{1}(ZG). In this situation one obtains Wall’s L^{s}-groups. □

The theory of projective L–groups fits into the scheme as follows: one defines 𝕃^{p}(A) = 𝕃^{h}(A^{∧}), where A^{∧} is the idempotent completion of A. The objects of A^{∧} are pairs (A, p) with A an object of A and p^{2} = p. The morphisms ϕ: (A, p) → (B, q) are the A-morphisms ϕ: A → B with qϕp = ϕ. Again it is possible to “partially” complete A. If K_{0}(A) ⊂ k ⊂ K_{0}(A^{∧}) is an involution invariant subgroup , we define A^{∧k} to be the full subcategory of A^{∧} with objects defining elements of k ⊂ K_{0}(A^{∧}). This way we may define 𝕃^{k}(A) = 𝕃(A^{∧k}). Similarly to the above, for k = K_{0}(A) ⊂ K_{0}(A^{∧}), L_{∗}^{k}(A) is naturally isomorphic to L_{∗}^{h}(A). The quotient K̃_{0}(A) = K_{0}(A^{∧})/K_{0}(A) is called the reduced projective class group of A.

Example 6.2. If A is the category of free ZG-modules then A^{∧} is isomorphic to the category of projective ZG-modules and L_{∗}^{p}(A) are Novikov’s original L^{p}- groups. □

Suppose M is a metric space with the finite group G acting by isometries, R a ring with involution. In [18] we defined an additive category C_{M,G}(R) with involution as follows:

Definition 6.3. An object A is a free left R(G)-module together with a map f: A ⟶ F (M), where F (M) is the set of finite subsets of M, satisfying

(1) f is G-equivariant.

(2) A_{x} = {a ∈ A | f(a) ⊆ {x}} is a finitely generated free sub R-module

(3) As an R-module A = ⨁_{x∈M} A_{x}

(4) f(a + b) ⊆ f(a) ∪ f(b)

(5) For each ball B ⊂ M, {x ∈ B | A_{x} ≠ 0} is finite. A morphism ϕ: A ⟶ B is a morphism of RG-modules, satisfying the following condition: there exists k so that the components ϕ_{n}^{m}: A_{m} ⟶ B_{n} (which are Rmodule morphisms) are zero when d(m, n) > k. C_{M,G}(R) is an additive category in an obvious way.

Given an object A, an R-module homomorphism ϕ: A ⟶ R is said to be locally finite if the set of x ∈ M for which ϕ(A_{x}) ≠ 0 is finite. We define A^{∗} = Hom_{R}^{lf}(A, R), the set of locally finite R-homomorphisms. We want to make ∗ a functor from C_{M,G}(R) to itself to make C_{M,G}(R) a category with involution. We define f^{∗}: A^{∗} ⟶ F M by f^{∗}(ϕ) = {x | ϕ(A_{x}) ≠ 0} which is finite by assumption. A^{∗} has an obvious right action of G turning it into a right RG module given by ϕg(a) = ϕ(ga), and f^{∗} is equivariant with respect to the right action on M given by xg = g^{−1}x. To make ∗ an endofunctor of C_{M,G}(R) we need to replace the right action by a left action. We do this by the standard way in surgery theory by letting g act on the left by letting g^{−1} act on the right. In the unoriented case, given a homomorphism w: G ⟶ {±1}, we let g act on the left of A^{∗} by w(g) · g^{−1} on the right.

Example 6.4. Let ρ_{W}: G → O(W) be an orthogonal action of G on a finite dimensional real vector space W. We take M = W with the action through ρ_{W}, and orientation character det(ρ_{W}). This will be called the standard orientation on C_{W,G}(Z). □

Remark 6.5. We will need to find a system of stable isomorphisms for the category C_{M,G}(R) to be able to do simple L–theory. To do this we choose a point x in each G-orbit, and an RG_{x}-basis for A_{x}, where G_{x} is the isotropy subgroup of x. We then extend that by equivariance to an R-basis of the module. Having a basis allows defining an isomorphism in the usual fashion. In each case we need to describe the indeterminacy in the choices coming from the choice of R-basis and points in the orbit. For our particular choices of M it will be easy to determine the subgroup s, so we will not formulate a general statement. □

We will study the L–theory of the categories C_{M,G}(R) using excision. Let N be a G-invariant metric subspace of M. Denoting C_{M,G}(R) by U, let A be the full subcategory on modules A so that A_{x} = 0 except for x in some bounded neighborhood of N. The category A is isomorphic to C_{N,G}(R) in an obvious fashion. The quotient category U/A, which we shall denote by C_{M,G}^{>N}(R), has the same objects as U but two morphisms are identified if the difference factors through A, or in other words, if they differ in a bounded neighborhood of N. This is a typical example of an additive category U which is A-filtered in the sense of Karoubi. We recall the definition.

Definition 6.6. Let A be a full subcategory of an additive category U. Denote objects of A by the letters A through F and objects of U by the letters U through W. We say that U is A-filtered, if every object U has a family of decompositions {U = E_{α} ⊕ U_{α}}, so that

(1) For each U, the decomposition form a filtered poset under the partial order that E_{α} ⊕ U_{α} ≤ E_{β} ⊕ U_{β}, whenever U_{β} ⊆ U_{α} and E_{α} ⊆ E_{β}.

(2) Every map A → U, factors A → E_{α} → E_{α} ⊕ U_{α} = U for some α.

(3) Every map U → A factors U = E_{α} ⊕ U_{α} → E_{α} → A for some α.

(4) For each U, V the filtration on U ⊕ V is equivalent to the sum of filtrations {U = E_{α} ⊕ U_{α}} and {V = F_{β} ⊕ V_{β}} i.e. to U ⊕ V = (E_{α} ⊕ F_{β}) ⊕ (U_{α} ⊕ V_{β})

The main excision results were proved in [26], [8], [9], [30]. We give a slight generalization of the L–theory results. Let 𝕂 denote the Quillen K–theory spectrum, and 𝕂^{−∞} its non-connective delooping (with the K_{−i}-groups as homotopy groups).

Theorem 6.7. Let U be an A-filtered additive category with involution. Consider the map i: K_{0}(A^{∧}) → K_{0}(U^{∧}) induced by inclusion, and let k = i^{−1}(K_{0}(U)). There are fibrations of spectra

𝕂(A^{∧k}) → 𝕂(U) → 𝕂(U/A)

and

𝕂^{−∞}(A) → 𝕂^{−∞}(U) → 𝕂^{−∞}(U/A)

If U and A admit compatible involutions there is a fibration of spectra

𝕃^{k}(A) → 𝕃^{h}(U) → 𝕃^{h}(U/A) .

More generally, if (i) a ⊂ K_{i}(A),b ⊂ K_{i}(U^{∧}), and c ⊂ K_{i}((U/A)^{∧}), for i ≦ 1, (ii) a = i^{−1}(b) and b → c is onto, (iii) a, b, and c contain K_{0}(A), K_{0}(U) and K_{0}(U/CA) respectively, if i = 0,

and (iv) a, b and c contain the indeterminacy subgroup given by the system of stable

isomorphisms in the case i = 1, then we have a fibration of spectra

𝕃^{a}(A) → 𝕃^{b}(U) → 𝕃^{c}(U/A)

Proof. The K–theory statements are implicitly contained in [26]. A simpler, more modern proof and explicit statements are given in [8]. The first L–theory statement was proved in [9], and the other L–theory statements follow by the following argument: we have an exact sequence

K_{0}(A^{∧k}) → K_{0}(U) → K_{0}(U/A) → 0,

where the map from K_{0}(U) → K_{0}(U/A) is onto because the categories have the same objects. Letting I denote the image of K_{1}(U/A) in K_{0}(A^{∧k}), we consider the diagram of short exact sequences:

[figure: 0; I; K_{0}(A^{∧k}); K_{0}(U); K_{0}(U/A); 0; 0; I; a; b′; K_{0}(U/A); 0; 0; I; a; b; c; 0; 0; I; K_{0}(A^{∧}); K_{0}(U^{∧}); K_{0}((U/A)^{∧})]

The vertical arrows are either equalities or inclusions. We define b′ simultaneously as the pullback of a/I → b → c and as the pushout of 0 → K_{0}(A^{∧k})/I → K_{0}(U) → K_{0}(U/A) → 0. We have

a/K_{0}(A^{∧k}) ≅ b′/K_{0}(U),

so using the Ranicki–Rothenberg fibrations of spectra [30]

\xymatrix{𝕃^{k}(A) \ar[r] \ar[d] & 𝕃^{a}(A) \ar[r] \ar[d] & ℍ(a/K_{0}(A^{∧k})) \ar[d] \\ 𝕃^{h}(U) \ar[r] & 𝕃^{b′}(U) \ar[r] & ℍ(b′/K_{0}(U))}

we get a fibration

𝕃^{a}(A) → 𝕃^{b′}(U) → 𝕃^{h}(U/A).

We now repeat this argument using the isomorphisms b/b′ ≅ c/K_{0}(U/A) to obtain the desired fibration of spectra. Since L^{h}-groups may be understood as simple L-groups with all of K_{1} as allowed torsions, the above bootstrapping argument extends to fibrations of the L-spectra stated, using the isomorphism

K_{1}(U/A)/ ker(∂) ≅ image(∂)

where ∂ is the boundary map δ: K_{1}(U/A) → K_{0}(A^{∧}). □

7. Bounded R_{−} transfers

Let G denote a finite group of even order, with a subgroup H < G of index 2. We first describe the connection between the bounded R_{−} transfer and the compact line bundle transfer of [38,12C] by means of the following diagram:

\xymatrix{L_{n}^{h}(ZG, w) \ar[d]_{_{trf_{R−}}} \ar[r]^{^{trf^{I−}}} & L_{n+1}^{h}(ZH →ZG, wϕ) \ar[d]^{j_{∗}} \\ L_{n+1}^{h}(C_{R_{−},G}(Z), wϕ) & L_{n+1}^{k,h}(ZH →ZG, wϕ) \ar[l]^{r_{∗}}}

where w: G → {±1} is the orientation character for G and ϕ: G → {±1} has kernel H. On C_{R_{−},G}(Z) we use the standard orientation defined in Example 6.4. Note that the orientation induced on C_{pt}(ZG) via the cone point inclusion c: C_{pt}(ZG) → C_{R_{−},G}(Z) is non–trivial. The homomorphism

r_{∗}: L_{n+1}^{k,h}(ZH →ZG, wϕ) → L_{n+1}^{h}(C_{R_{−},G}(Z), wϕ)

is obtained by adding a ray [1, ∞) to each point of the boundary double cover in domain and range of a surgery problem. Here k in the decoration means that we are allowing projective ZH-modules that become free when induced up to ZG

Theorem 7.1. The map r_{∗}: L_{n+1}^{k,h}(ZH → ZG, wϕ) → L_{n+1}^{h}(C_{R_{−},G}(Z), wϕ) is an isomorphism, and under this identification, the bounded R_{−} transfer corresponds to the line bundle transfer, followed by the relaxation of projectivity map j_{∗} given by k.

Proof. Let A be the full subcategory of U = C_{R_{−},G}(Z) with objects that are only nontrivial in a bounded neighborhood of 0. Then C_{R_{−},G}(Z) is A-filtered. The category A is equivalent to the category of free ZG-modules (with the non-orientable involution). The quotient category U/A is equivalent to C_{[0,∞),H}^{>0}(Z), which has the same L-theory as C_{R}(ZH), so we get a fibration of spectra

𝕃^{k}(ZH) → 𝕃^{h}(ZG) → 𝕃^{h}(C_{R_{−},G}(Z)) .

This shows that

𝕃^{h}(C_{R_{−},G}(Z)) ≃ 𝕃^{k,h}(ZH → ZG) □

The line bundle transfer can be studied by the long exact sequence

(7.2) ⋯ → LN_{n}(ZH →ZG, wϕ) → L_{n}^{h}(ZG, w) → L_{n+1}^{h}(ZH →ZG, wϕ)

→ LN_{n−1}(ZH →ZG, wϕ) → L_{n−1}^{h}(ZG, w) → …

given in [38,11.6]. The obstruction groups LN_{n}(ZH →ZG, wϕ) for codimension 1 surgery have an algebraic description

(7.3)  LN_{n}(ZH →ZG, wϕ) ≅ L_{n}^{h}(ZH, α, u)

given by [38,12.9]. The groups on the right–hand side are the algebraic L–groups of the “twisted” anti–structure defined by choosing some element t ∈ G − H and then setting α(x) = w(x)t^{−1}x^{−1}t for all x ∈ H, and u = w(t)t^{−2}. Another choice of t ∈ G−H gives a scale equivalent anti–structure on ZH. The same formulas also give a “twisted” anti–structure (ZG, α, u) on ZG, but since the conjugation by t is now an inner automorphism of G this is scale equivalent to the standard structure (ZG, w). We can therefore define the twisted induction map

ĩ_{∗}: L_{n}^{h}(ZH, α, u) → L_{n}^{h}(ZG, w)

and the twisted restriction map

γ̃_{∗}: L_{n}^{h}(ZG, w) → L_{n}^{h}(ZH, α, u)

as the composites of the ordinary induction or restriction maps (induced by the inclusion (ZH, w) → (ZG, w)) with the scale isomorphism.

The twisted anti–structure on ZH is an example of a “geometric anti–structure” [21,p.110]:

α(g) = w(g)θ(g^{−1}), u = ±b

where θ: G → G is a group automorphism with θ^{2}(g) = bgb^{−1}, w ∘ θ = w, w(b) = 1 and θ(b) = b.

Example 7.4. For G cyclic, the orientation character restricted to H is trivial, θ(g) = tgt^{−1} = g and u = w(t)t^{2}. Choosing t ∈ G a generator we get b = t^{2}, which is a generator for H. □

There is an identification [14,Thm.3], [22,50-53] of the exact sequence (7.2) for the line bundle transfer, extending the scaling isomorphism L_{n}^{h}(ZG, w) ≅ L_{n}^{h}(ZG, α, u) and (7.3), with the long exact sequence of the “twisted” inclusion

… L_{n}^{h}(ZH, α, u)⟶^{ĩ_{∗}}L_{n}^{h}(ZG, α, u) → L_{n}^{h}(ZH →ZG, α, u) → L_{n−1}^{h}(ZH, α, u) → …

These identifications can then be substituted into the following “twisting diagram” in order to compute the various maps (see [22,Appendix 2] for a complete tabulation in the case of finite 2–groups).

\xymatrix{ & LN_{n}(ZH →ZG, wϕ) \ar[rr]^{ĩ_{∗}} \ar[dr] &  & L_{n}(ZG, w) \ar[rr]^{γ_{∗}} \ar[dr] &  & L_{n}(ZH, w) \\ (7.5) &  & L_{n+1}(γ_{∗}) \ar[ur] \ar[dr] &  & L_{n+1}(ZH →ZG, wϕ) \ar[ur] \ar[dr] \\  & L_{n+1}(ZH, w) \ar[ur] \ar[rr]_{i_{∗}} &  & L_{n+1}(ZG, wϕ) \ar[ur] \ar[rr]_{γ̃_{∗}} &  & LN_{n−1}(ZH →ZG, wϕ)}

The existence of the diagram depends on the identifications L_{n+1}(γ_{∗}) ≅ L_{n}(γ̃_{∗}) and L_{n+1}(i_{∗}) ≅ L_{n}(ĩ_{∗}) obtained geometrically in [14] and algebraically in [28].

In Section 16 we need to use bounded surgery groups with geometric anti– structure (see [18]). The new ingredient is a counterpart to the automorphism θ: H → H at the metric space level. The necessary definitions for (C_{M,H}(R), α, u), for M a metric space on which H acts by isometries and R a ring, were given in Section 6.

Let θ_{H}: H ⟶ H be a group automorphism so that the data (H, θ_{H}, w, b) gives a geometric anti–structure on RH. Let θ_{M}: M ⟶ M be an isometry with the properties θ_{M}(g · m) = θ_{H}(g) · θ_{M}(m), θ_{M}^{2}(m) = bm, and θ_{H}^{2}(g) = bgb^{−1}.

Given an object A ∈ G_{M,H} (R) , we have the functor ∗ from G_{M,H} (R) to itself so that G_{M,H} (R) is a category with involution. We may then twist the involution ∗ by composing with the functor sending (A, f) to (A^{θ}, f^{θ}) where A^{θ} is the same R-module, but g acts on the left by multiplication by θ(g) and f^{θ} = θ_{M}^{−1} · f. This defines the bounded anti–structure on G_{M,H} (R) and on the subcategory C_{M,H}(R) of free RH modules.

Example 7.6. Bounded geometric anti–structures arise geometrically as above. The bounded R_{−} transfer sits in the long exact sequence

LN_{n}(C_{W ×R_{−},G}(Z), wϕ) → L_{n}^{h}(C_{W,G}(Z), w) → L_{n+1}^{h}(C_{W ×R_{−},G}(Z), wϕ)

→ LN_{n−1}(C_{W ×R_{−},G}(Z), wϕ) → L_{n−1}^{h}(C_{W,G}(Z), w) → …

where w = det(ρ_{W}) is the standard orientation (see Example 6.4). The bounded LN–group

LN_{n}(C_{W ×R_{−},G}(Z), wϕ) ≅ L_{n}(C_{W,H}(Z), α, u)

where θ_{W} (x) = t · x and θ_{H}(h) = tht^{−1} for a fixed t ∈ G − H.

Conversely, given a bounded geometric antistructure (θ_{H}, θ_{M}, b, w), we can define G = ⟨H, t | t^{−1}ht = θ_{H}(h), t^{2} = b⟩ and t · m = θ_{M}(m). Then C_{M,G}(R) induces (θ_{H}, θ_{M}, b, w) as above, showing that all geometric antistructures arise by twisting and restricting to an index two subgroup. □

The L-theory of these bounded geometric anti–structures also has a useful vanishing property which we now wish to formulate. We first give a basic construction.

Definition 7.7. If A is an additive category, then the opposite category A^{op} is the category with the same objects as A but hom_{A^{op}}(A, B) = hom_{A}(B, A). The product category A × A^{op} is an additive category with involution given by ∗: (A, B) = (B, A) on objects and ∗: (α, β) = (β, α) on morphisms.

Clearly K_{i}(A^{op}) = K_{i}(A), so we can identify K_{i}(A × A^{op}) = K_{i}(A) × K_{i}(A).

Lemma 7.8. Let b ⊆ K_{i}(A) for some i ≦ 1, and q = b × b ⊆ K_{i}(A × A^{op}). Then L_{n}^{q}(A × A^{op}) = 0 for all n.

Proof. Let P(A) denote the category with the same objects as A, but with morphisms given by A–isomorphisms. Then it suffices to prove that the quadratic category Q(A × A^{op}) ≃ P(A) via the hyperbolic map (see [39,p.122]). This shows that L_{∗}^{h}(A × A^{op}) = 0 and other decorations follow trivially from the Ranicki Rothenberg exact sequences (note that the Tate cohomology H^{∗}(q) = 0). The result for lower L–groups follows by replacing A by C_{R}(A).

Suppose (ν_{1}, ν_{2}): (A, B) → (B, A) is a non–singular quadratic form representing an element in Q(A × A^{op}). This means that the bilinearization ν_{1} + ν_{2} is an isomorphism, and we are allowed to change (ν_{1}, ν_{2}) by terms of the form (α, β) − (β, α). We have

(ν_{1}, ν_{2}) + (ν_{2}, 0) − (0, ν_{2}) = (ν_{1} + ν_{2}, 0)

and the right hand side is a hyperbolic form. □

We encounter the A × A^{op} situation in the following setting:

Example 7.9. Let M = M_{1} ∪ M_{2} be a metric space given as the union of two sub–metric spaces M_{1} and M_{2}, where we denote M_{1} ∩ M_{2} by N. Suppose that

(i) G acts by isometries on M, such that each g ∈ G preserves or switches M_{1} and M_{2} in this decomposition,

(ii) H = {g ∈ G | g(M_{1}) = M_{1}} is an index two subgroup of G, (iii) for every k > 0 there exists an l > 0 such that, if x ∈ M_{1} (resp. x ∈ M_{2})

with d(x, N) > l, then d(x, M_{2}) > k (resp. d(x, M_{1}) > k). The category C_{M,H}(R) has a bounded geometric antistructure (α, u) given by (θ_{H}, θ_{M}, b, w) as in Example 7.6, with θ_{M}(m) = t · m and θ_{H}(h) = tht^{−1} for a fixed t ∈ G − H. Next, observe that the category

C_{M,H}^{>N}(R) = C_{M_{1},H}^{>N}(R) × C_{M_{2},H}^{>N}(R)

because of our separation condition (ii). Moreover, the functor T: C_{M_{1},H}^{>N}(R) → C_{M_{2},H}^{>N}(R)^{op} defined by T (A, f) = (A^{∗}, θ_{M}^{−1} ∘ f^{∗}) on objects and T (ϕ) = ϕ^{∗} on morphisms is an equivalence of categories. We are thus in the A × A^{op} situation described above and L_{∗}^{h}(C_{M,H}^{>N}(R), α, u) = 0 by Lemma 7.8. □

For any bounded geometric antistructure, notice that the action of θ_{M} on M takes H–orbits to H–orbits since θ_{M}(g · m) = θ_{H}(g) · θ_{M}(m). Let M_{(H,θ)} denote the subset consisting of H–orbits in M which are fixed by the θ–action. Then M_{(H,θ)} = {m ∈ M | θ_{M}(m) ∈ H · m}. Note that M_{(H,θ)} is a H–invariant subspace of M.

Theorem 7.10. Suppose that (C_{M,H}(R), α, u) has a bounded geometric antistructure (α, u) given by (θ_{H}, θ_{M}, b, w), such that:

(i) M = O(K), where K is a finite H–CW complex and M has the cone of the given H–action on K,

(ii) θ_{M} is induced by a simplicial map on K, (iii) M_{(H,θ)} ⊆ O(L) := N for some H–invariant subcomplex L ⊂ K, and (iv) for some i ≦ 1, I ⊂ K_{i}(C_{M,H}^{>N} (R)) is a subgroup with H^{∗}(I) = 0.

Then L_{n}^{I}(C_{M,H}^{>N} (R), α, u) = 0 for all n.

Corollary 7.11. Let G = C(2^{r}q), q odd, be a cyclic group and H ⊂ G the subgroup of index 2. Let W be a G–representation, and N = ⋃_{K}{W^{K} | [G : K] is odd}. Then L_{n}^{I}(C_{W,H}^{>N} (Z), α, u)(q) = 0 on the top component, where (α, u) is the antistructure given above.

Proof of Theorem 7.10. We extend the given H–action on M to a simplicial action of G = ⟨H, t | t^{−1}ht = θ_{H}(h), t^{2} = b⟩ as described above. The proof is by induction on cells, so suppose that K is obtained from L by attaching exactly one G–equivariant k–cell D^{k} × G/G_{0}. Since M_{(H,θ)} ⊂ O(L) = N, it follows that G_{0} ⊂ H and we may write G/G_{0} = H/G_{0} ⊔ tH/G_{0}. Now we define M_{1} = O(L ∪ (D^{k} × H/G_{0})) and consider the category A = C_{M_{1},H}^{>O(L)}(R). By construction, we have

C_{M,H}^{>N}(R) = A × A^{op}

which has trivial L–theory by Lemma 7.8. Since the Tate cohomology of the K_{1} decoration I vanishes, we get L_{n}^{I}(C_{M,H}^{>N} (R), α, u) = 0. □

8. Calculations in bounded K-theory

We begin to compute the bounded transfers trf_{W} by considering the bounded K–theory analogue. In this section, G = C(2^{r}q) is cyclic of order 2^{r}q, with r ≧ 2 and q ≧ 1 odd. By Theorem 3.8 we can restrict our attention to those W where the isotropy subgroups have 2–power index. Let G_{i} ⊆ G denote the subgroup of index [G : G_{i}] = 2^{i} for i = 0, 1, …, r. As above, we reserve the notation H < G for the subgroup of index 2.

Any real, orthogonal G–representation W can be decomposed uniquely into isotypical direct summands indexed by the subgroups K ⊆ G, where in each summand G operates with isotropy group K away from the origin. Since we assume that W has isotropy of 2–power index, we can write

W = W [0] ⊕ W [1] ⊕ ⋯ ⊕ W [r]

where W [i] is isotypic with isotropy group G_{i}. Thus W [0] = R^{k} is a trivial G– representation, and W [1] is a sum of R_{−} factors. We note that W is complex if dim W [1] is even. If W is complex, then W_{max} ⊆ W denotes a 2-dimensional complex sub-representation with maximal proper isotropy subgroup. Then W_{max} is either irreducible or W_{max} = R_{−} ⊕ R_{−}.

We study bounded K–theory by means of equivariant filtrations of the control space. The basic sequence is (see[18]):

⋯ → K_{i+1}(C_{V,G}^{>U}(Z))⟶^{∂_{i+1}}K_{i}(C_{U,G}(Z)) → K_{i}(C_{V,G}(Z)) → K_{i}(C_{V,G}^{>U}(Z)) → …

valid for U ⊆ V a closed G–invariant subspace. If W_{1} is a complex representation with dim W_{1} = 2 and isotropy group K ≠ G, let U = ⋃ ℓ_{α} be the union of [G : K] rays from the origin in W_{1}, which are freely permuted by G/K. Then W_{1} \ ⋃ ℓ_{α} is a disjoint union of open fundamental domains for the free G/K–action.

If W = W_{1} ⊕W_{2}, we call W_{2} ⊂ ⋃ ℓ_{α} ×W_{2} ⊂ W the orbit type filtration of W.

Lemma 8.1. Let W = W_{1} ⊕ W_{2}, where W_{1} is a complex 2–dimensional subrepresentation of W with minimal isotropy subgroup K ≠ G. Then

K_{i+1}(C_{W,G}^{>⋃ ℓ_{α}×W_{2}}(Z)) ≅ K_{i−1−k}(ZK)
K_{i}(C_{⋃ ℓ_{α}×W_{2},G}^{>W_{2}}(Z)) ≅ K_{i−1−k}(ZK),

where k = dim W_{2}. The boundary map ∂_{i+1} = 1 − t_{∗} in the long exact sequence of the orbit type filtration for W.

Proof. The bounded category C_{⋃ ℓ_{α}×W_{2},G}^{>W_{2}}(Z) of germs away from W_{2} has effective fundamental group K, as defined in [18,3.13]. It therefore has the same K–theory as C_{R^{k+1}}(C_{pt}(ZK)). The other case is similar. □

Since K_{−j}(ZK) = 0 for j ≧ 2 by [10], this Lemma gives vanishing results for bounded K–theory as well.

Lemma 8.2. Suppose that W is complex, and W^{G} = 0. Then the inclusion map induces an isomorphism

K_{i}(C_{W_{max}×R_{−},G}(Z)) → K_{i}(C_{W ×R_{−},G}(Z))

for i ≦ 1.

Proof. This is an argument using the orbit type filtration. Let W = W_{max} ⊕ W_{2}, and suppose that W_{2} ≠ 0 or equivalently dim W_{2} ≧ 2, since W is complex. Write W = W′ ⊕ W′′ with W′ ⊆ W_{2}, dim W′ = 2 and Iso(W′) = K minimal. We choose W′′ containing W_{max}, and by induction we assume the result holds for W′′.

Then applying Lemma 8.1, we get the calculations

(8.3)  K_{i}(C_{⋃ ℓ_{α}×W′′×R_{−},G}^{>W′′×R_{−}}(Z)) = K_{i−2−|W′′|}(ZK)

and

(8.4)  K_{i}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×W′′×R_{−}}(Z)) = K_{i−3−|W′′|}(ZK) .

Since dim W′′ ≧ 2, we get K_{i}(C_{⋃ ℓ_{α}×W′′×R_{−},G}^{>W′′×R_{−}}(Z)) = 0 for i ≦ 2 by [10], and

K_{i}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×W′′×R_{−}}(Z)) = 0 for i ≦ 3. From the exact sequence of the filtration, it

follows that K_{i}(C_{W ×R_{−},G}^{>W′′×R_{−}}(Z)) = 0 for i ≦ 2, and therefore

K_{i}(C_{W′′×R_{−},G}(Z))⟶^{≈}K_{i}(C_{W ×R_{−},G}(Z))

for i ≦ 1. We are done, by induction. □

Corollary 8.5. K_{i}(C_{W ×R_{−},G}^{>W_{max}×R_{−}}(Z)) = 0 for i ≦ 2.

Proof. We continue the notation from above, and look at part of the filtration sequence

K_{i}(C_{W′′×R_{−},G}^{>W_{max}×R_{−}}(Z)) → K_{i}(C_{W ×R_{−},G}^{>W_{max}×R_{−}}(Z)) → K_{i}(C_{W ×R_{−},G}^{>W_{max}×R_{−}}(Z)) .

The first term is zero for i ≦ 2 by induction on dimension, and the third term is zero for i ≦ 2 as above. □

We can obtain a little sharper result with some additional work. First a useful observation:

Lemma 8.6. Let A be an additive category (with involution). Then the map R → R sending x to −x induces minus the identity on K-theory (and L-theory) of C_{R}(A).

Proof. The category C_{R}(A) is filtered by the full subcategory whose objects have support in a bounded neighborhood of 0. This subcategory is equivalent to A and the quotient category my be identified with C_{[0,∞)}^{>0}(A) × C_{(−∞,0]}^{>0}(A) via the projection maps in an obvious way.

Consider the diagram

[figure: A; C_{R}(A); C_{[0,∞)}^{>0}(A) × C_{(−∞,0]}^{>0}(A); A; C_{[0,∞)}(A); C_{[0,∞)}^{>0}]

where the vertical map is induced by x ↦ |x|. In the lower horizontal row, K and L–theory of the middle term is trivial, so the boundary map will be an isomorphism. The lower row splits off the upper row in two different ways, one induced by including [0, ∞) ⊂ R and the other by sending x ∈ [0, ∞) to −x ∈ R. Under these two splittings we may identify K or L–theory of the quotient C_{[0,∞)}^{>0}(A)×C_{(−∞,0]}^{>0}(A) with C_{[0,∞)}^{>0} × C_{[0,∞)}^{>0} and under this identification, the flip map of R corresponds to interchanging the two factors. On K–theory (or L–theory) we conclude that the exact sequence is of the form

0 → A_{∗} → A_{∗} × A_{∗}⟶^{+}A_{∗} → 0 .

The flip action on the last term is trivial, and on the middle term it interchanges the two factors, so the inclusion must send a to (a, −a). Hence the flip action on the first term must be a ↦ −a. □

Let t_{∗} denote the induced action of a generator t ∈ G on K–theory, and ε: G → {±1} the action of G on R_{−}.

Proposition 8.7. Let W be a complex 2-dimensional G–representation with W^{G} = 0, and isotropy subgroup K. Then under the isomorphisms of (8.3) and (8.4) the complex

K_{i+1}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×R_{−}}(Z))⟶^{∂_{a}}K_{i}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z))⟶^{∂_{b}}K_{i−1}(C_{R_{−},G}^{>0}(Z))

with ∂_{b} ∘ ∂_{a} = 0 is isomorphic to

K_{i−2}(ZK)⟶^{∂′_{a}}K_{i−2}(ZK)⟶^{∂′_{b}}K_{i−2}(ZH)

where ∂′_{a} = 1 − ε(t)t_{∗} = 2 and ∂′_{b} = Ind_{H} ∘(1 − t_{∗}) = 0.

Proof. The orbit type filtration is based on a G–equivariant simplicial model for W, where G acts through the projection to G/K. The third term in the complex is K_{i−1}(C_{R_{−},G}^{>0}(Z)) = K_{i−2}(ZH) and the identification of the boundary maps follows from the definition of the germ categories.

To compute ∂_{a}, we use the isomorphisms between the domain of ∂_{a} and K_{i−2}(ZK) obtained by noticing that every element is induced from an element of

K_{i+1}(C_{C×R,K}^{>∂C×R}(Z)),

where C is the region between two adjacent half–lines of ⋃ ℓ_{α}. This follows since the regions in the complement of ⋃ ℓ_{α} × R_{−} are disjoint and the boundedness condition ensures there is no interference. Similarly the isomorphism of the range of ∂_{a} with K_{i−2}(Z) is obtained by noticing that every element is induced from C_{h×R,K}^{>R} where h is just one half–line in ⋃ ℓ_{α}, and we can think of ∂C = h ∪ th. To compute the boundary we first take the standard boundary to C_{∂C×R,K}(Z) which is an isomorphism, and then map away from 0 × R. It follows from the proof of Lemma 8.6 above that this map is of the form a ↦ (a, −a) in K or L–theory. In this picture, the support of one of the boundary components is along h and the other along th. We need to use the group action to associate both elements to the same ray. Since t flips the R–factor, we get a change of sign before adding, so ∂_{a} sends a to 2a.

To compute ∂_{b}, we start with an element in the source of ∂_{b} which as above is identified via induction with K_{i}(C_{h×R,K,G}^{>R}(Z)). The boundary first sends this isomorphically to C_{R,K}(Z), then by induction to C_{R_{−},G}(Z), and then via the natural map to the range of ∂_{b}, which is C_{R_{−},G}^{>0}. We have a commutative square

\xymatrix{K_{1}(C_{h×R,K}^{>R}(Z)) \ar[r]^{Ind} \ar[d]_{∂} & K_{1}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) \ar[d]^{∂_{b}} \\ K_{0}(C_{R,K}^{>0}(Z)) \ar[r]^{Ind} & K_{0}(C_{R_{−},G}^{>0}(Z))}

where C_{R,K}^{>0}(Z) = C_{R,K}(Z) × C_{R,K}(Z). Under this identification, the natural map K_{0}(C_{R,K}(Z)) → K_{0}(C_{R,K}^{>0}(Z)) is just a ↦ (a, −a), and the induction map

Ind: K_{0}(C_{R,K}^{>0}(Z)) → K_{0}(C_{R_{−},G}^{>0}(Z))

is given by (a, b) ↦ Ind_{H}(a + t_{∗}b). In this case the action of t on C_{R_{−},G}(Z) is the identity since any element is invariant under the action of G, hence under the action of t. It follows that Ind(a, −a) = Ind_{H}(a − t_{∗}a) = 0 as required. □

Lemma 8.8. Let W be a complex 2-dimensional G–representation with proper isotropy group K. The composition

K_{i+1}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) → K_{i+1}(C_{W ×R_{−},G}^{>R_{−}}(Z)) → K_{i}(C_{R_{−},G}(Z))

is zero for i ≦ 1.

Proof. By Proposition 8.7 the given composite map factors through the cone point inclusion K_{i}(ZG) → K_{i}(C_{R_{−},G}(Z)). For i = 1, the result follows from the fact that K_{2}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z) = K_{0}(ZK) is torsion, except for the factor K_{0}(Z) = Z, and

Wh(ZG) is torsion free. For i = 0, we use the surjection K_{0}(Q̂K) → K_{−1}(ZK), and compute with Q̂ coefficients and i = 1. Note that the given composite is just the boundary map ∂: K_{i+1}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) → K_{i}(C_{R_{−},G}(Z)).

We will list the steps, and leave the details to the reader. First, compute that K_{1}(C_{R_{−},G}(Q̂)) = K_{1}(Q̂G)/K_{1}(Q̂H) surjects onto K_{1}(C_{⋃ ℓ_{α}×R_{−},G}(Q̂)), by means of a braid containing the cone point inclusions into C_{R_{−},G}(Q̂) and C_{⋃ ℓ_{α}×R_{−},G}(Q̂). Second, prove that K_{1}(C_{⋃ ℓ_{α}×R_{−},G}(Q̂)) fits into a short exact sequence

0 → K_{1}(C_{⋃ ℓ_{α},H}(Q̂)) → K_{1}(C_{⋃ ℓ_{α},G}(Q̂)) → K_{1}(C_{⋃ ℓ_{α}×R_{−},G}(Q̂)) → 0

by means of a braid containing the inclusion ⋃ ℓ_{α} ⊆ ⋃ ℓ_{α} × R_{−}. Finally, compute K_{1}(C_{⋃ ℓ_{α},G}(Q̂)) = K_{1}(Q̂G)/K_{1}(Q̂K), and K_{1}(C_{⋃ ℓ_{α},H}(Q̂)) = K_{1}(Q̂H)/K_{1}(Q̂K) by comparing the groups under the inclusion H < G. We conclude that

K_{1}(C_{R_{−},G}(Q̂)) ≅ K_{1}(C_{⋃ ℓ_{α}×R_{−},G}(Q̂))

under the inclusion map, and hence ∂ = 0. □

Corollary 8.9. Let W be a complex G–representation with W^{G} = 0. Then the inclusion induces an isomorphism K_{i}(C_{R_{−},G}(Z)) → K_{i}(C_{W ×R_{−},G}(Z)) for i ≦ 0, and an injection for i = 1. If K_{−1}(ZK) = 0 for the maximal proper isotropy groups K of W, then K_{1}(C_{R_{−},G}(Z)) ≅ K_{1}(C_{W ×R_{−},G}(Z)).

Proof. We may assume that dim W = 2, and apply the filtering argument again. By (8.3) and (8.4) we get K_{i}(C_{W ×R_{−},G}^{>R_{−}}(Z)) = 0 for i ≦ 0, and K_{1}(C_{W ×R_{−},G}^{>R_{−}}(Z)) is

a quotient of K_{1}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) = K_{−1}(ZK). The boundary map

∂: K_{1}(C_{W ×R_{−},G}^{>R_{−}}(Z)) → K_{0}(C_{R_{−},G}(Z))

composed with the natural map K_{0}(C_{R_{−},G}(Z)) → K_{0}(C_{R_{−},G}^{>0}(Z)) is just the map ∂_{b} of Proposition 8.7. Since ∂_{b} = 0 on K–theory, the image Im ∂ factors through the inclusion K_{0}(ZH) → K_{0}(ZG) (because K ⊆ H), hence vanishes in K_{0}(C_{R_{−},G}(Z)) by Lemma 8.8.

Similarly, K_{2}(C_{W ×R_{−},G}^{>R_{−}}(Z)) is a quotient of K_{2}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) = K_{0}(ZK), be-

cause the boundary map K_{2}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×R_{−}}(Z)) = K_{−1}(ZK) to K_{1}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) = K_{−1}(ZK) is multiplication by 2, and hence injective. Then we make the same argument, based on Lemma 8.8.

If we also assume K_{−1}(ZK) = 0, then K_{1}(C_{W ×R_{−},G}^{>R_{−}}(Z)) = 0 so we get the isomorphism K_{1}(C_{R_{−},G}(Z))⟶^{≈}K_{1}(C_{W ×R_{−},G}(Z)). □

9. The double coboundary

The composite Δ of maps from the L^{h} − L^{p} and L^{s} − L^{h} Rothenberg sequences

\xymatrix{H^{i}(K̃_{0}(ZG)) \ar[r] \ar[dr]^{Δ} & L_{i}^{h}(ZG) \ar[d] \\  & H^{i}(Wh(ZG))}

(see [20]) has an algebraic description by means of a “double coboundary” homomorphism

δ^{2}: H^{i}(K̃_{0}(ZG)) → H^{i}(Wh(ZG))

In this section, we will give a brief description due to Ranicki [32] of this homomorphism (see also [27,§6.2] for related material on “interlocking” exact sequences in K and L–theory).

Let X be a space with a Z/2 action T: X → X, and define homomorphisms

Δ: H^{i}(π_{n}(X)) → H^{i}(π_{n+1}(X))

by sending g: S^{n} → X to

h ∪ (−1)^{i}T h: S^{n+1} = D_{+}^{n+1} ∪_{S^{n}} D_{−}^{n+1} → X

for any null-homotopy h: D^{n+1} → X of the map g + (−1)^{i+1}T g: S^{n} → X.

The maps Δ lead to a universal description of double coboundary maps, as follows. Let f: X → Y be a Z/2–equivariant map of spaces with Z/2 action, and consider the long exact sequence

⋯ → π_{n}(X)⟶^{f}π_{n}(Y) → π_{n}(f) → π_{n−1}(X) → π_{n−1}(Y) → …

We define I_{n} = ker(f: π_{n}(X) → π_{n}(Y)) and J_{n} = Im(π_{n}(Y) → π_{n}(f)), and get an exact sequence

0 → π_{n}(X)/I_{n} → π_{n}(Y) → π_{n}(f) → I_{n−1} → 0

which can be spliced together from the short exact sequences

(9.1)  \begin{matrix}0 → π_{n}(X)/I_{n} → π_{n}(Y) → J_{n} → 0 \\ 0 → J_{n} → π_{n}(f) → I_{n−1} → 0 . \end{matrix}

Then it follows directly from the definitions that the double coboundary

δ^{2}: H^{i}(I_{n−1})⟶^{δ}H^{i+1}(J_{n})⟶^{δ}H^{i}(π_{n}(X)/I_{n})

from the Tate cohomology sequences induced by (9.1) is given by the composite

δ^{2}: H^{i}(I_{n−1})⟶^{inc_{∗}}H^{i}(π_{n−1}(X))⟶^{Δ}H^{i}(π_{n}(X))⟶^{proj_{∗}}H^{i}(π_{n}(X)/I_{n}) .

If we can pick the map f: X → Y appropriately, say with I_{n} = 0 and I_{n−1} = π_{n−1}(X), this gives an algebraic description of Δ. We give some examples, with G denoting a finite cyclic group as usual.

Example 9.2. There is an exact sequence [20]

0 → Wh(ZG) → Wh(Ẑ_{2}G) → Wh(ZG→Ẑ_{2}G) → K̃_{0}(ZG) → 0

of Z/2 modules and the associated double coboundary in Tate cohomology equals

Δ: H^{i}(K̃_{0}(ZG)) → H^{i}(Wh(ZG)) .

The point here is that ker(Wh(ZG) → Wh(Ẑ_{2}G)) = 0 [25], and the map K̃_{0}(ZG) → K̃_{0}(Ẑ_{2}G) is zero by a result of Swan [36]. We could also use the exact sequence

0 → Wh(ZG) → Wh(ẐG) ⊕ Wh(QG) → Wh(Q̂G)) → K̃_{0}(ZG) → 0

to compute the same map Δ. □

Example 9.3. There is an exact sequence

0 → Wh(C_{R_{−},G}(Z)) → Wh(C_{R_{−}}(Ẑ ⊕ Q)) →

Wh(C_{R_{−}}(Q̂)) → K̃_{0}(C_{R_{−},G}(Z)) → 0

where Wh(C_{R_{−},G}(Z)) = Wh(ZH → ZG^{−}), K̃_{0}(C_{R_{−},G}(Z)) = K̃_{0}(ZH → ZG^{−}), and ZG^{−} denotes the group ring ZG with the non–oriented involution induced by t → −t^{−1}. The point here is that K_{2}(Q̂G) = K_{2}(C_{R_{−}}(Q̂)) maps trivially through K_{1}(ZG) into Wh(C_{R_{−},G}(Z)) because SK_{1}(ZG) = 0 [25]. We therefore get an algebraic description of δ^{2}: H^{i}(K̃_{0}(ZH → ZG^{−})) → H^{i}(Wh(ZH → ZG^{−})) as used in the statement of Theorem B. □

Example 9.4. There is an exact sequence

0 → Wh(C_{W ×R_{−},G}(Z)) → Wh(C_{W ×R_{−}}(Ẑ ⊕ Q)) →

Wh(C_{W ×R_{−}}(Q̂)) → K̃_{0}(C_{W ×R_{−},G}(Z)) → 0

for any complex G–representation W with W^{G} = 0. We therefore get an algebraic description of δ^{2}: H^{i}(K̃_{0}(C_{W ×R_{−},G}(Z))) → H^{i}(Wh(C_{W ×R_{−},G}(Z))) as used in the statement of Theorem C. □

Lemma 9.5. For complex G–representations W_{1} ⊆ W with W^{G} = 0, there is a commutative diagram

\xymatrix{H^{i}(K̃_{0}(C_{W_{1}×R_{−},G}(Z))) \ar[d]_{c_{∗}} \ar[r]^{δ^{2}} & H^{i}(Wh(C_{W_{1}×R_{−},G}(Z))) \ar[d]^{c_{∗}} \\ H^{i}(K̃_{0}(C_{W ×R_{−},G}(Z))) \ar[r]^{δ^{2}} & H^{i}(Wh(C_{W ×R_{−},G}(Z)))}

where the vertical maps are induced by the inclusion W_{1} ⊆ W.

For our applications, the main point of the double coboundary description is that it permits these maps induced by cone point inclusions to be computed using bounded K–theory, instead of bounded L–theory.

The double coboundary maps also commute with restriction to subgroups of G.

Proposition 9.6. Let G_{1} < G be a subgroup of odd index, and H_{1} < G_{1} have index 2, then there are twisted restriction maps H^{i}(K̃_{0}(ZH → ZG^{−})) → H^{i}(K_{1}(ZH_{1} → ZG_{1}^{−})) such that the diagram

\xymatrix{H^{i}(K̃_{0}(ZH →ZG^{−})) \ar[r]^{δ^{2}} \ar[d]_{Res} & H^{i}(K_{1}(ZH →ZG^{−})) \ar[d]^{Res} \\ H^{i}(K̃_{0}(ZH_{1}→ZG_{1}^{−})) \ar[r]^{δ_{1}^{2}} & H^{i}(K_{1}(ZH_{1}→ZG_{1}^{−}))}

commutes.

Proof. The vertical maps are twisted restriction maps given by the composite of the twisting isomorphisms H^{i}(K̃_{0}(ZH → ZG^{−})) ≅ H^{i+1}(K̃_{0}(ZH → ZG)) and H^{i}(K_{1}((ZH →ZG^{−})) ≅ H^{i+1}(K_{1}((ZH →ZG)) with the restriction maps induced by the inclusion ZG_{1} → ZG of rings with involution. Since G_{1} < G has odd index, H_{1} = H ∩ G_{1} and the composition Res_{G_{1}} ∘ Ind_{H} lands in the image of Ind_{H_{1}} by the double coset formula. □

This can be generalized to the double coboundary maps used in the statement of Theorem C, under certain conditions.

Proposition 9.7. Let G_{1} < G be an odd index subgroup, and H_{1} < G_{1} have index 2. Suppose that W only has proper isotropy subgroups of 2–power index. Then there are a twisted restriction maps H^{i}(K̃_{0}(C_{W ×R_{−},G}(Z))) → H^{i}(K̃_{0}(C_{Res W ×R_{−},G_{1}}(Z))) and H^{i}(Wh(C_{W ×R_{−},G}(Z))) → H^{i}(Wh(C_{Res W ×R_{−},G_{1}}(Z))) which commute with the corresponding double coboundary maps δ_{W}^{2} and δ_{Res W}^{2} .

10. Some basic facts in K- and L-theory

In this section we record various calculational facts from the literature about K– and L–theory of cyclic groups. A general reference for the K–theory is [25], and for L–theory computations is [20]. Recall that K̃_{0}(A) = K_{0}(A^{∧})/K_{0}(A) for any additive category A, and Wh(A) is the quotient of K_{1}(A) by the subgroup defined by the system of stable isomorphisms.

Theorem 10.1. Let G be a cyclic group, K a subgroup. We then have (1) K_{1}(ZG) = (ZG)^{∗} ⊂ K_{1}(QG) Here (ZG)^{∗} denotes the units of ZG.

(2) The torsion in K_{1}(ZG)) is precisely {±G}, so Wh(ZG) is torsion free.

(3) The maps K_{1}(ZK) → K_{1}(ZG) and

Wh(ZG)/ Wh(ZK) → Wh(QG)/ Wh(QK)

are injective.

(4) K̃_{0}(ZG) a torsion group and the map K̃_{0}(ZG) → K̃_{0}(Z_{(p)}G is the zero map for all primes p.

(5) K_{−1}(ZG) is torsion free, and sits in an exact sequence

0 → K_{0}(Z) → K_{0}(ẐG) ⊕ K_{0}(QG) → K_{0}(Q̂G) → K_{−1}(ZG) → 0 .

(6) K_{−1}(ZK) → K_{−1}(ZG) is an injection.

(7) K_{−j}(ZG) = 0 for j ≧ 2.

Proof. The proof mainly consists of references. See [25,pp.6,14] for the first two parts. Part (3) follows from (1) and the relation (ZG)^{∗} ∩ (QK)^{∗} = (ZK)^{∗}. Part (4) is due to Swan [36], and part (7) is a result of Carter [10]. Part (5) gives the arithmetic sequence for computing K_{−1}(ZG), and the assertion that K_{−1}(ZG) is torsion free is easy to deduce (see also [10]). Since Res_{K} ∘ Ind_{K} is multiplication by the index [G : K], part (6) follows from (5). □

Tate cohomology of K_{i}-groups play an important role. The involution on K– theory is induced by duality on modules. It is conventionally chosen to have the boundary map

K_{1}(Q̂(G) → K̃_{0}(ZG)

preserve the involution, so to make this happen we choose to have the involution on K_{0} be given by sending [P] to −[P^{∗}], and the involution on K_{1} is given by sending τ to τ^{∗}. This causes a shift in dimension in Ranicki-Rothenberg exact sequences

… → H^{0}(K̃_{0}(A)) → L_{2k}^{h}(A) → L_{2k}^{p}(A) → …

compared to

… → H^{1}(Wh(A)) → L_{2k}^{s}(A) → L_{2k}^{h}(A) → …

Theorem 10.2. Let G be a cyclic group, K a subgroup.

(1) L_{2k}^{s}(ZG), L_{2k}^{p}(ZG), and L2k^{⟨−1⟩}(ZG) are torsion–free when k is even, and when k is odd the only torsion is a Z/2-summand generated by the Arf invariant element.

(2) The groups L_{2k+1}^{h}(ZG) = L_{2k+1}^{s}(ZG) = L_{2k+1}^{p}(ZG) are zero (k even), or Z/2 (k odd).

(3) L2k+1^{⟨−1⟩}(ZG) = H^{1}(K_{−1}(ZG)) (k even), or Z/2 ⊕ H^{1}(K_{−1}(ZG)) (k odd).

(4) The Ranicki-Rothenberg exact sequence gives

0 → H^{0}(K̃_{0}(ZG)) → L_{2k}^{h}(ZG) → L_{2k}^{p}(ZG) → H^{1}(K̃_{0}(ZG)) → 0

so L_{2k}^{h}(ZG) has the torsion subgroup H^{0}(K̃_{0}(ZG)).

(5) The double coboundary δ^{2}: H^{1}(K̃_{0}(ZG)) → H^{0}(Wh(ZG)) is injective:

(6) The maps L_{2k}^{s}(ZK) → L_{2k}^{s}(ZG), L_{2k}^{p}(ZK) → L_{2k}^{p}(ZG) , and L2k^{⟨−1⟩}(ZK) →

L2k^{⟨−1⟩}(ZG) are injective when k is even. For k odd and [G : K] even, the kernel is generated by the Arf invariant element.

(7) In the oriented case, Wh(ZG) has trivial involution and H^{1}(Wh(ZG)) = 0.

Proof. See [20] for the proof of part (1). Part (2) is due to Bak for L^{s} and L^{h} [2]. Part (7) is due to Wall [25] and part(6) follows again from Res_{K} ∘ Ind_{K} = [G : K]. The other parts, follow from substituting in the change of K–theory sequence. □

11. The computation of L_{1}^{p}(ZG, w)

Here we correct an error in the statement of [16,5.1] (Notice however that Table 2 [16,p.553] has the correct answer).

Proposition 11.1. Let G = σ × ρ, where σ is an abelian 2–group and ρ has odd order. Then L_{n}^{p}(ZG, w) = L_{n}^{p}(Zσ, w) ⊕ L_{n}^{p}(Zσ→ZG, w) where w: G → {±1} is an orientation character. For i = 2k, the second summand is free abelian and detected by signatures at the type U(C) representations of G which are non–trivial on ρ. For n = 2k + 1, the second summand is a direct sum of Z/2’s, one for each type U(GL) representation of G which is non–trivial on ρ.

Remark 11.2. Note that type U(C) representations of G exist only when w ≡ 1, and type U(GL) representations of G exist only when w ≢ 1. In both cases, the second summand is computed by transfer to cyclic subquotients of order 2^{r}q, q > 1 odd, with r ≧ 2. □

Proof. The given direct sum decomposition follows from the existence of a retraction of the inclusion σ → G compatible with w. It also follows that

L_{n+1}^{p,h}(ZG→Ẑ_{2}G, w) ≅ L_{n+1}^{p,h}(Zσ→Ẑ_{2}σ, w) ⊕ L_{n}^{p}(Zσ→ZG, w)

since the map L_{n}^{h}(Ẑ_{2}σ, w) → L_{n}^{h}(Ẑ_{2}G, w) is an isomorphism. The computation of the relative groups for Z → Ẑ_{2} can be read off from [16,Table 2,Remark 2.14]: for each centre field E of a type U(GL) representation, the contribution is H^{0}(C(E)) ≅ Z/2 if i ≡ 1 mod 2.

The detection of L_{i}^{p}(Zσ→ZG, w) by cyclic subquotients is proved in [21, 1.B.7, 3.A.6, 3.B.2]. □

Corollary 11.3. Let G = C(2^{r}q), for q odd and r ≧ 2. If q = 1 assume that r ≧ 3. Then the group

L_{1}^{p}(ZG, w)(q) = ⨁_{i=2}^{r} CL_{2}^{K}(E_{i}) ≅ (Z/2)^{r−1}

when w ≢ 1, where the summand CL_{2}^{K}(E_{i}) = H^{0}(C(E_{i})), 2 ≦ i ≦ r, corresponds to the rational representation with centre field E_{i} = Q(ζ_{2^{i}q}).

Remark 11.4. The calculation of L_{1}^{p} contradicts the assertion in [7,p.733,l.-8] that the projection map G → C(2^{r}) induces an isomorphism on L_{1}^{p} in the non–oriented case. This error means that there is a gap in the proofs of the main results of [7] for cyclic groups not of 2–power order. □

We now return to our main calculational device for determining non–linear similarities of cyclic groups, namely the “double coboundary”

δ^{2}: H^{1}(K̃_{0}(ZG)) → H^{1}(W h(ZG))

from the exact sequence

0 → W h(ZG) → W h(ẐG) ⊕ K_{1}(QG) → K_{1}(Q̂G) → K̃_{0}(ZG) → 0 .

We recall that the discriminant induces an isomorphism

L_{1}^{h}(ZG, w) ≅ H^{1}(W h(ZG), w)

since L_{i}^{s}(ZG, w) ≅ L′_{i}(ZG, w) = 0 for i ≡ 0, 1 mod 4 by the calculations of [38,3.4.5,5.4].

Proposition 11.5. The kernel of the map L_{1}^{h}(ZG, w) → L_{1}^{p}(ZG, w) equals the image of the double coboundary δ^{2}: H^{1}(K̃_{0}(ZG)) → H^{1}(W h(ZG)) under the isomorphism L_{1}^{h}(ZG, w) ≅ H^{1}(W h(ZG), w) induced by the discriminant .

Proof. We will use the commutative braid

\xymatrix{ & H^{1}(K̃_{0}(ZG)) \ar[rr]^{δ^{2}} \ar[dr] &  & H^{1}(W h(ZG)) \ar[rr] \ar[dr] &  & L_{0}^{s}(ZG, w) \\ (11.6) &  & L_{1}^{h}(ZG, w) \ar@{-->}[ur] \ar[dr] &  & H^{1}(Δ) \ar[ur] \ar[dr] \\  & L_{1}^{s}(ZG, w) \ar[ur] \ar[rr] &  & L_{1}^{p}(ZG, w) \ar[ur] \ar[rr] &  & H^{0}(K̃_{0}(ZG))}

relating the L^{h} to L^{p} and the L^{s} to L^{h} Rothenberg sequences. The term H^{1}(Δ) is the Tate cohomology of the relative group for the double coboundary. We see that the discriminant of an element σ ∈ L_{1}^{h}(ZG, w) lies in the image of the double coboundary if and only if σ ∈ ker(L_{1}^{h}(ZG, w) → L_{1}^{p}(ZG, w)). □

Corollary 11.7. There is an isomorphism L_{1}^{p}(ZG, w) ≅ H^{1}(Δ).

Remark 11.8. There is a version of these results for L_{3}^{p}(ZG, w) as well, on the kernel of the projection map L_{3}^{p}(ZG, w) → L_{3}^{p}(ZK, w), where K = C(4). The point is that L_{i}^{s}(ZG, w) ≅ L_{i}^{s}(ZK, w) is an isomorphism for i ≡ 2, 3 ( mod 4) as well [38,3.4.5,5.4]. There is also a corresponding braid (15.1) for L_{2k+1}^{s}(C_{W ×R_{−},G}(Z)), L_{2k+1}^{h}(C_{W ×R_{−},G}(Z)) and L_{2k+1}^{p}(C_{W ×R_{−},G}(Z)) involving the double coboundary in bounded K–theory. The cone point inclusion

C_{pt}(ZG, w) = C_{pt,G^{−}}(Z) → C_{W ×R_{−},G}(Z)

induces a natural transformation between the two braid diagrams. □

In Section 12 we will need the following calculation. We denote by L_{n}^{Wh(ZH)}(ZG^{−}) the L–group of ZG with the non-oriented involution, and Whitehead torsions allowed in the subgroup Wh(ZH) ⊂ Wh(ZG).

Lemma 11.9. L_{1}^{Wh(ZH)}(ZG^{−}) = 0, and the discriminant map L_{0}^{Wh(ZH)}(ZG^{−}) → H^{0}(Wh(ZH) is an injection.

Proof. The Rothenberg sequence gives

L_{n}^{s}(ZG^{−}) → L_{n}^{Wh(ZH)}(ZG^{−}) → H^{n}(Wh(ZH)) .

For n ≡ 1 (mod 4) the outside terms are zero, and hence L_{1}^{Wh(ZH)}(ZG^{−}) = 0. For n ≡ 0 (mod 4), L_{0}^{s}(ZG^{−}) = 0 as noted above and the injectivity follows. □

Another specific calculation needed later is:

Lemma 11.10. For G = C(2^{r}) and H < G of index 2, the group L_{1}^{k}(ZH) = 0, where k = ker(K̃_{0}(ZH) → K̃_{0}(ZG)).

Proof. The group L_{1}^{k}(ZH) = 0 sits in the arithmetic sequence [20]

⋯ → L_{2}^{k,h}(ZH →Ẑ_{2}H) → L_{1}^{k}(ZH) → L_{1}^{h}(Ẑ_{2}H) → …

and L_{1}^{h}(Ẑ_{2}H) = 0 since H is a 2–group. The relative group L_{2}^{k,h}(ZH → Ẑ_{2}H) ≅ L_{2}^{k̄,h}(M → M̂_{2}), where M denotes an involution invariant maximal order in QH containing ZH, and k̄ is the preimage in K_{1}(M → M̂_{2}) of k. By the tables in [20,Appendix], we see that L_{2}^{k̃,h}(M → M̂_{2}) ≅ H^{0}(k̃), so it remains to compute this group. However, K_{1}(M → M̂_{2}) was computed in [17,4.8] as a module over Z/2. It is the direct sum of copies of Ẑ_{2} with −1 action of the involution, and one copy of Z[Z/2] generated by 1 + 2i in each centre field Q(ζ_{2^{k}}), for k ≧ 2. It is not difficult to check using 2–adic units 1 + 2t^{2^{k}} (as in [17,§6]) that the elements 1 + 2i in each centre field are trivial in K̃_{0}(ZH). Therefore k̄ has involution −1, and H^{0}(k̄) = 0. □

In later sections, it will be convenient to stabilize with trivial representations and use the identification

L_{n+k}^{p}(C_{W ⊕R^{k},G}(Z)(Z)) ≅ Ln^{⟨−k⟩}(C_{W,G}(Z)(Z)).

The composite with the transfer

trf_{R^{k}}: L_{n}^{p}(ZG) → L_{n+k}^{p}(C_{W ⊕R^{k},G}(Z)(Z))

is just the usual “change of K–theory” map, which may be analysed by the Ranicki-Rothenberg sequences [31]. For G a finite group, K_{−j}(ZG) = 0 if j ≧ 2 so only the first stablization is needed.

Lemma 11.11. For G = C(2^{r}q) and w: G → {±1} non–trivial, the map

L_{2k+1}^{p}(ZG, w) → L2k+1^{⟨−1⟩}(ZG, w)

is injective.

Proof. The group K_{−1}(ZG) is a torsion free quotient of K_{0}(Q̂G), and has the involution induced by [P] ↦ −[P^{∗}] on K_{0}(Q̂G) [16,3.10]. This implies first that H^{0}(K_{0}(Q̂G)) = 0, and so the image of the coboundary

H^{0}(K_{−1}(ZG)) → H^{1}(K_{0}(ẐG)) ⊕ H^{1}(K_{0}(QG))

consists of the classes (0, [E]) where E splits at every prime. We need to compare the exact sequences in the following diagram (see [16], [20]):

\xymatrix{ &  & L_{2k+2}^{K}(ẐG, w) ⊕ L_{2k+2}^{K}(QG, w) \ar[r] \ar[d] & L_{2k+2}^{K}(Q̂G, w) \ar[d] \\ 0 \ar[r] & L2k+2^{⟨−1⟩}(ZG, w) \ar[r] \ar[d] & L_{2k+2}^{p}(ẐG, w) ⊕ L_{2k+2}^{p}(QG, w) \ar[r] \ar[d] & L_{2k+2}^{p}(Q̂G, w) \ar[d] \\ 0 \ar[r] & H^{0}(K_{−1}(ZG)) \ar[r] & H^{1}(K_{0}(ẐG)) ⊕ H^{1}(K_{0}(QG)) \ar[r] & H^{1}(K_{0}(Q̂G))}

The groups L_{2k+2}^{K}(ẐG, w) reduce to the L–groups of finite fields, which are zero in type U. For each involution invariant field E in the top component of QG, the group L_{2k+2}^{K}(E) = H^{0}(E^{×}) which maps injectively into L_{2k+2}^{K}(Ê) = H^{0}(Ê^{×}), [16]. It follows that the images of L_{2}^{K}(E) and L2k+2^{⟨−1⟩}(ZG, w) in L_{2k+2}^{p}(E) have zero intersection , and so the composite map L2k+2^{⟨−1⟩}(ZG, w) → H^{1}(K_{0}(QG)) is an isomorphism. Therefore L2k+2^{⟨−1⟩}(ZG, w) → H^{0}(K_{−1}(ZG)) is surjective and we conclude that the map L_{2k+1}^{p}(ZG, w) → L2k+1^{⟨−1⟩}(ZG, w) is injective. □

Corollary 11.12. Let G = C(2^{r}q) and w: G → {±1} the non–trivial orientation. If W is a G–representation with W^{G} = 0, then the map

L_{2k+1}^{p}(C_{W,G}(Z)(Z), w) → L2k+1^{⟨−1⟩}(C_{W,G}(Z)(Z), w)

is injective.

Proof. We first note that the cone point maps K_{0}(RG) → K_{0}(C_{W,G}(R)) are surjective for R = Ẑ, Q or Q̂ since for these coefficients RG has vanishing K_{−1} groups. This shows that K_{−1}(C_{W,G}(Z)) is again a quotient of K_{0}(Q̂G). To see that K_{−1}(C_{W,G}(Z)) is also torsion free, consider the boundary map K_{1}(C_{W,G}^{>0}(Q̂)) → K_{0}(C_{W} (Q̂)) which is just a sum of induction maps K_{0}(Q̂K) → K_{0}(Q̂G) from proper subgroups of K ⊂ G. But for G cyclic, these induction maps are split injective. We now complete the argument by comparing the diagram above with the corresponding diagram for the bounded theory, concluding that H^{0}(K_{−1}(ZG)) → H^{0}(K_{−1}(C_{W,G}(Z))) is surjective. Since L2k+2^{⟨−1⟩}(ZG, w) → H^{0}(K_{−1}(ZG)) is also surjective, we are done. □

12. The proof of Theorem A

The condition (i) is equivalent to assuming that S(V_{1}) and S(V_{2}) are freely G– homotopy equivalent. Condition (ii) is necessary by Corollary 3.10 which rules out non–linear similarities of semifree representations. Condition (ii) also implies that S(V_{1}) is s–normally cobordant to S(V_{2}) by [3,Prop.2.1], which is another necessary condition for topological similarity. Thus under conditions (i) and (ii), there exists a homotopy equivalence f: S(V_{2}) → S(V_{1}), and an element σ = σ(f) ∈ L_{0}^{h}(ZG) such that trf_{R_{−}}(σ) = 0 ∈ L_{1}^{h}(C_{R_{−},G}(Z)) if and only if V_{1} ⊕ R_{−} ∼_{t} V_{2} ⊕ R_{−}.

Comparing the h- and s- surgery exact sequences it is easy to see that the image of σ in H^{0}(Wh(ZG)) is given by the Whitehead torsion {τ(f)} = {Δ(V_{1})/Δ(V_{2})} ∈ Wh(ZG) of the homotopy equivalence S(V_{2})/G ≃ S(V_{1})/G.

In Section 2 we gave the short exact sequence

0 → Wh(ZG)/ Wh(ZH) → Wh(ZH →ZG) → k → 0

where K_{1}(C_{R_{−},G}(Z))/{±G}) is denoted by Wh(C_{R_{−},G}(Z)) = Wh(ZH → ZG) and k = ker(K̃_{0}(ZH) → K̃_{0}(ZG)). We proved in Theorem 3.6 that the transfer of the torsion element in Wh(ZG) in Wh(ZH →ZG) is given by the same element under the map induced by inclusion Wh(ZG) → Wh(ZH →ZG).

It follows that the image of trf_{W} (σ) in H^{1}(Wh(C_{R_{−},G}(Z))) = H^{1}(Wh(ZH → ZG^{−})) is given by the image of our well-defined element

{Δ(V_{1})/Δ(V_{2})} ∈ H^{1}(Wh(ZG^{−})/ Wh(ZH))

under the cone point inclusion into H^{1}(Wh(ZH →ZG^{−})).

The necessity of the condition is now easy. To have a non-linear similarity we must have

trf_{R_{−}}(σ) = 0 ∈ L_{1}^{h}(C_{R_{−},G}(Z)) .

Hence

trf_{R_{−}}(Δ(V_{1})/Δ(V_{2})) = 0 ∈ H^{1}(Wh(ZH →ZG^{−}))

must vanish by naturality of the transfer in the Rothenberg sequence. This element comes from H^{1}(Wh(ZG^{−})/ Wh(ZH)), so to vanish in H^{1}(Wh(ZH →ZG^{−})) it must be in the image from

H^{0}(k) → H^{1}(Wh(ZG^{−}/ Wh(ZH))

under the double coboundary.

To prove sufficiency, we assume that the image of the transferred element

{Δ(V_{1})/Δ(V_{2})} ∈ H^{1}(Wh(ZH →ZG^{−}))

is zero. Consider the long exact sequence derived from the inclusion of filtered categories

C_{pt}(ZG) ⊂ C_{R_{−},G}(Z)

where C_{R_{−},G}(Z) has the standard orientation (6.4) inducing the non–trivial orientation at the cone point. The quotient category C_{R_{−},G}^{>0}(Z) of germs away from 0 is canonically isomorphic to C_{[0,∞),H}^{>0}(Z) , since by equivariance what happens on the positive half line has to be copied on the negative half line, and what happens near 0 does not matter in the germ category. Since the action of H on [0, ∞) is trivial, this category is precisely C_{[0,∞)}^{>0}(ZH) which has the same K- and L-theory as C_{R}(ZH) by the projection map. By Theorem 6.7 we thus get a long exact sequence

… → L_{n}^{k}(ZH) → L_{n}^{h}(ZG^{−}) → L_{n}^{h}(C_{R_{−},G}(Z)) → L_{n−1}^{k}(ZH) → …

where the map is induced by induction. Comparing this sequence with the long exact sequence for the pair L_{n}^{h,k}(ZH → ZG^{−}) (see Theorem 7.1) it follows that

L_{n}^{h}(C_{R_{−},G}(Z)) ≅ L_{n}^{k,h}(ZH → ZG^{−}) .

Consider the following diagram with exact rows and columns:

\xymatrix{L_{1}^{Wh(ZH)}(ZG^{−}) \ar[r] \ar[d] & L_{1}^{h,Wh(ZH)}(ZH →ZG^{−}) \ar[r] \ar[d] & L_{0}^{h}(ZH) \ar[d] \\ L_{1}^{h}(ZG^{−}) \ar[r] \ar[d] & L_{1}^{k,h}(ZH →ZG^{−}) \ar[d] \ar[r] & L_{0}^{k}(ZH) \ar[d] \\ H^{1}(Wh(G^{−})/ Wh(H)) \ar[r] & H^{1}(Wh(ZH →ZG^{−})) \ar[r] & H^{1}(k)}

We need to show that trf_{R_{−}}(σ) vanishes in order to produce the non-linear similarity. We know that the image of Res_{H}(σ) = 0, and our assumption is that the image

{Δ(V_{1})/Δ(V_{2})} ∈ H^{1}(Wh(ZH →ZG^{−}))

vanishes. We will finish the argument by showing:

Lemma 12.1. Suppose that σ ∈ L_{0}^{h}(ZG).

(i) L_{1}^{h,Wh(ZH)}(ZH →ZG^{−}) is torsion–free, and (ii) the torsion subgroup of L_{1}^{k,h}(ZH →ZG^{−}) injects into H^{1}(Wh(ZH →ZG^{−})). (iii) If Res_{H}(σ) = 0 ∈ L_{0}^{h}(ZH), then trf_{R_{−}}(σ) is a torsion element.

Proof of Lemma 12.1. For assertion (i), we consider the diagram

[figure: L_{0}^{s}(ZH); L_{0}^{s}(ZG^{−}); L_{0}^{h}(ZH); L_{0}^{Wh(ZH)}(ZG^{−}); H^{0}(Wh(ZH)); H^{0}(Wh(ZH))]

where L_{0}^{s}(ZG^{−}) = 0. Since L_{1}^{Wh(ZH)}(ZG^{−}) = 0 as well (by Lemma 11.9), it follows that

L_{1}^{h,Wh(ZH)}(ZH →ZG^{−}) = ker(L_{0}^{h}(ZH) → L_{0}^{Wh(ZH)}(ZG^{−})) ≅ L_{0}^{s}(ZH)

where L_{0}^{s}(ZH) is torsion–free. Part (ii) follows from part (i), since the previous term in the exact sequence has exponent two. For assertion (iii) we refer to part of the twisting diagram of Section 7, namely the commutative diagram:

[figure: L_{0}^{h}(ZG); _{trf_{R_{−}}}; _{Res_{H}}; L_{1}^{h}(ZG^{−}); L_{1}^{k,h}(ZH →ZG^{−}); L_{0}^{k}(ZH)]

and the fact that Res_{H}(trf_{R_{−}}(σ)) = 0 since it factors through Res_{H}: L_{0}^{h}(ZG) → L_{0}^{h}(ZH). But L_{1}^{h}(ZG^{−}) ≅ H^{1}(Wh(ZG^{−}) has exponent 2. □

Remark 12.2. Theorem A gives necessary and sufficient conditions for the existence of 5–dimensional similarities. Consider the situation in dimensions ≦ 5. By character theory it suffices to consider a cyclic group G, which must be of order divisable by 4, say 4q, with index 2 subgroup H. It suffices by Lemma 3.1 to consider the following situation

V_{1} ⊕ W ∼_{t} V_{2} ⊕ W

where V_{i} are free homotopy equivalent representations which become isomorphic once restricted to H. Therefore S(V_{1}) is normally cobordant to S(V_{2}). If V_{i} are two-dimensional they are determined by one character, so homotopy equivalence implies isomorphism. We may thus assume V_{i} are at least 4-dimensional, and dim W = 1. According to Theorem 3.5, non-linear similarity is now determined by trf_{W} (σ), where σ is an element of infinite order in L_{0}^{h}(ZG) hitting the element in the structure set determined by the homotopy equivalence of S(V_{2})/G ≃ S(V_{1})/G. In case W is the trivial representation we may identify trf_{W} with the map

L_{0}^{h}(ZG) → L_{0}^{p}(ZG)

which we have seen (in Corollary 3.10) is injective on the elements of infinite order. Hence we are left with the case where W is the non-trivial one-dimensional representation R_{−}, which is reduced to number theory by Theorem A. In Corollary 19.2 we work out the number theory for G = C(2^{r}), but the general case remains open. □

13. The proof of Theorem B

The new ingredient in Theorem B is the double coboundary. As in the last section, we may assume that f: S(V_{2}) → S(V_{1}) is a G–homotopy equivalence which is freely G–normally cobordant to the identity, giving an element σ = σ(f) ∈ L_{0}^{h}(ZG). Then V_{1} ⊕ R_{−} ⊕ R_{+} ∼_{t} V_{2} ⊕ R_{−} ⊕ R_{+} if and only if

trf_{R_{−}⊕R_{+}}(σ) = 0 ∈ L_{2}^{h}(C_{R_{−}⊕R_{+},G}(Z))

by Theorem 3.5. But L_{2}^{h}(C_{R_{−}⊕R_{+},G}(Z)) = L_{1}^{p}(C_{R_{−},G}(Z)), so we may regard the criterion as the vanishing of trf_{R_{−}}(σ) ∈ L_{1}^{p}(C_{R_{−},G}(Z)) instead.

The main commutative diagram is:

\xymatrix{ & H^{1}(K̃_{0}(C_{R_{−},G}(Z))) \ar[rr]^{δ^{2}} \ar[dr] &  & H^{1}(K_{1}(C_{R_{−},G}(Z)) \ar[rr] \ar[dr] &  & L_{0}^{s}(C_{R_{−},G}(Z)) \\ (13.1) &  & L_{1}^{h}(C_{R_{−},G}(Z)) \ar@{-->}[ur] \ar[dr] &  & H^{1}(Δ_{R_{−}}) \ar@{-->}[ur] \ar[dr] \\  & L_{1}^{s}(C_{R_{−},G}(Z)) \ar[ur] \ar[rr] &  & L_{1}^{p}(C_{R_{−},G}(Z)) \ar[ur] \ar[rr] &  & H^{0}(K̃_{0}(C_{R_{−},G}(Z)))}

where H^{1}(Δ_{R_{−}}) denotes the relative group of the double coboundary map. Notice that some of the groups in this diagram already appeared as relative L–groups in the last section. We have

L_{1}^{h}(C_{R_{−},G}(Z)) = L_{1}^{k,h}(ZH →ZG^{−})

and

L_{1}^{s}(C_{R_{−},G}(Z)) = L_{1}^{h,Wh(ZH)}(ZH →ZG^{−}) .

By Lemma 12.1(ii) we again conclude that the torsion subgroup of L_{1}^{p}(C_{R_{−},G}(Z)) injects into H^{1}(Δ_{R_{−}}).

However, since Res_{H}(σ) = 0 ∈ L_{0}^{h}(ZH), Lemma 12.1(i) implies that trf_{R_{−}}(σ) is a torsion element. Furthermore, its image in H^{1}(Δ_{R_{−}}) is zero if and only if

{Δ(V_{1})/Δ(V_{2})} ∈ H^{1}(Wh(ZH →ZG^{−}))

lies in the image of the double coboundary, and this completes the proof. □

14. Calculations in bounded L–theory

Suppose that σ = σ(f) ∈ L_{0}^{h}(ZG) is the surgery obstruction arising from a normal cobordism between S(V_{1}) and S(V_{2}), as in the statement of Theorem 3.5. In this section, we establish two important properties of trf_{W} (σ) in preparation for the proof of Theorem C. Unless otherwise mentioned, all bounded categories will have the standard orientation (see Example 6.4).

For a complex G–representation W the standard orientation is trivial, and the cone point inclusion 0 ∈ W induces the map

c_{∗}: L_{n}^{h}(ZG) → L_{n}^{h}(C_{W,G}(Z))

Note that the presence of an R_{−} factor introduces a non-trivial orientation at the cone point

c_{∗}: L_{n+1}^{h}(ZG, w) → L_{n+1}^{h}(C_{W ×R_{−},G}(Z))

where w: G → {±1} is the non-trivial projection. The properties are:

Theorem 14.1. Suppose W is a complex G–representation with no R_{+} summands. If Res_{H}(V_{1} ⊕ W) ⊕ R_{+} ∼_{t} Res_{H}(V_{2} ⊕ W) ⊕ R_{+}, then

(i) trf_{W ×R_{−}}(σ) ∈ L_{2k+1}^{h}(C_{W ×R_{−},G}(Z)) is a torsion element, and (ii) there exists a torsion element σ̂ ∈ L_{2k+1}^{p}(ZG, w) such that

c_{∗}(σ̂) = trf_{W ⊕R_{−}}(σ) ∈ L_{2k+1}^{p}(C_{W ×R_{−},G}(Z))

where dim W = 2k.

Before giving the proof, we need some preliminary results. When the L–theory decoration is not explicitly given, we mean L^{⟨−∞⟩}.

Lemma 14.2. Let W be a complex G–representation. Then

L_{2k+1}(C_{W,G}(Z)) ⊗ Q = 0

for k ≧ 0

Proof. We argue by induction on dim W, starting with

L_{2k+1}(ZG) = Z/2 ⊕ H^{1}(K_{−1}(ZG))

which is all 2–torsion. It is enough to prove the result for the top component L_{2k+1}(C_{W,G}(Z))(q), and therefore by Theorem 3.8 we may assume that the isotropy groups of W all have 2–power index. Since we are working with L^{⟨−∞⟩}, we may ignore R_{+} summands of W. In the proof, all L–groups will be tensored with Q.

Let W = W′ ⊕ W′′ where dim W′ = 2 and W′ has minimal isotropy group K. We assume the result for W′′, and let ⋃ ℓ_{α} ⊂ W′ be a G-invariant set of rays from the origin, dividing W′ into fundamental domains for the free G/K-action.

Then

L_{n}(C_{⋃ ℓ_{α}×W′′,G}^{>W′′} (Z)) = L_{n−1}(C_{W′′,G}(Z)) = L_{n−1−|W′′|}(ZK),

which vanishes for n even, and

L_{n}(C_{W,G}^{>⋃ ℓ_{α}×W′′} (Z)) = L_{n−2}(C_{W′′,G}(Z)) = L_{n−2−|W′′|}(ZK),

which vanishes for n odd. Moreover, we have a long exact sequence

⋯ → L_{n}(C_{⋃ ℓ_{α}×W′′,G}^{>W′′} (Z)) → L_{n}(C_{W,G}^{>W′′} (Z)) → L_{n}(C_{W,G}^{>⋃ ℓ_{α}×W′′} (Z)) → …

We claim that the first map in this sequence is injective. The previous map in the long exact sequence is

L_{n+1}(C_{W,G}^{>⋃ ℓ_{α}×W′′} (Z)) ⟶^{∂} L_{n}(C_{⋃ ℓ_{α}×W′′,G}^{>W′′} (Z)),

which may be identified (using the isomorphisms above and Proposition 8.7) with a geometrically induced map

L_{n−1−|W′′|}(ZK)⟶^{1−u}L_{n−1−|W′′|}(ZK)

“multiplication by 1 − w(t)”, where t̄ = w(t)t^{−1} from the action of t in the antistructure used to define the L-group.

In the oriented case, w(t) = +1 and this boundary map is zero. Therefore, we conclude that

L_{2k+1}(C_{⋃ ℓ_{α}×W′′,G}^{>W′′} (Z)) ⟶^{≈} L_{2k+1}(C_{W,G}^{>W′′} (Z))

is an isomorphism and so

L_{2k+1}(C_{W,G}^{>W′′} (Z)) = L_{2k−|W′′|}(ZK) .

Finally, we will substitute this computation and our inductive assumption into the long exact sequence

⋯ → L_{2k+1}(C_{W′′,G}(Z)) → L_{2k+1}(C_{W,G}(Z)) →

L_{2k+1}(C_{W,G}^{>W′′} (Z)) → L_{2k}(C_{W′′,G}(Z)) →

and find that

0 → L_{2k+1}(C_{W,G}(Z)) → L_{2k+1}(C_{W,G}^{>W′′} (Z)) → L_{2k}(C_{W′′,G}(Z)).

However, the second map in this sequence can be identified with the inclusion map

Ind_{K}: L_{2k}(C_{W′′,K}(Z)) → L_{2k}(C_{W′′,G}(Z))

and the composition Res_{K} ∘ Ind_{K} is multiplication by [G : K], which is a rational isomorphism. Therefore Ind_{K} is injective and L_{2k+1}(C_{W,G}(Z)) = 0. □

Another computation we will need is Lemma 14.3. L_{n}(C_{W ×R_{−},G}^{>R_{−}}(Z)) ⊗ Q = 0 for W a complex G-representation.

Proof. We may assume that W^{G} = 0 and argue by induction on the dimension of W. We write W = W′ ⊕ W′′, where Iso(W′) = K is minimal (2–power index isotropy subgroups may be assumed as usual). We have 2 long exact sequences (all L–groups are tensored with Q):

(i) from the inclusion W′′ ⊕ R_{−} ⊂ W ⊕ R_{−}. For short, let A = C_{W ×R_{−},G}^{>R_{−}}(Z),

B = C_{W′′×R_{−},G}^{>R_{−}}(Z), and then A/B = C_{W ×R_{−},G}^{>W′′×R_{−}}(Z). We need the piece of the L-group sequence:

(14.4)  ... → L_{n}(B) → L_{n}(A) → L_{n}(A/B)...

and note that L_{n}(B) = 0 by induction.

(ii) we study L_{n}(A/B) by looking at the usual rays ⋃ ℓ_{α} ⊂ W′ which divide the 2-dim representation W’ into G/K chambers. Let

D_{0} = C_{⋃ ℓ_{α}×W′′×R_{−},G}^{>W′′×R_{−}}(Z)

and

D_{1} = C_{W ×R_{−},G}^{>⋃ ℓ_{α}×W′′×R_{−}}(Z)

Then we need the L–group sequence

(14.5)  → L_{n}(D_{0}) → L_{n}(A/B) → L_{n}(D_{1}) → L_{n−1}(D_{0})

But the groups L_{n}(D_{i}) are the ones we have been computing by using the chamber structure. In particular,

L_{n}(D_{0}) = L_{n−2}(C_{W′′,K}(Z))

and

L_{n}(D_{1}) = L_{n−3}(C_{W′′,K}(Z)).

But since K is the minimal isotropy group in W, it acts trivially on W′′ and these L-groups are just

L_{n}(D_{0}) = L_{n−2−|W′′|}(ZK)

and

L_{n}(D_{1}) = L_{n−3−|W′′|}(ZK).

Now the boundary map

(14.6)  L_{n}(D_{1}) → L_{n−1}(D_{0})

in the sequence is just multiplication by 1 − w(t) = 2 (w(t) = −1 since R_{−} is nonoriented), and this is a rational isomorphism. Therefore, L_{n}(A/B) = 0 by (14.5) and substituting back into (14.4), the L_{n}(A) = 0. □

Now a more precise result in a special case:

Lemma 14.7. Let W be a complex 2–dimensional G-representation with W^{G} = 0. Then L_{3}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)) = 0

Proof. We have an exact sequence

⋯ → L_{3}^{s}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) → L_{3}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)) →

L_{3}^{I}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×R_{−}}(Z)) → L_{2}^{s}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) → …

arising from the orbit type filtration. Here

I = Im(K_{2}(C_{W ×R_{−},G}^{>R_{−}}(Z)) → K_{2}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×R_{−}}(Z)))

= ker(K_{2}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×R_{−}}(Z)) → K_{1}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)))

= ker(K_{−1}(ZK)⟶^{2}K_{−1}(ZK))

by Proposition 8.7, where K is the isotropy subgroup of W. But K_{−1}(ZK) is torsion–free, so I = {0} ⊆ K_{2}. Now substitute the computations

L_{3}^{s}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) = L_{1}^{p}(ZK) = 0
L_{3}^{I}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×R_{−}}(Z)) = L_{0}^{p}(ZK)
L_{2}^{s}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) = L_{0}^{p}(ZK)

into the exact sequence. The boundary map

L_{3}^{I}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×R_{−}}(Z)) → L_{2}^{s}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z))

is multiplication by 2 so L_{3}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)) = 0. □

Remark 14.8. The same method shows that L_{1}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)) = L_{3}^{p}(ZK) = Z/2 for dim W = 2 as above, assuming that K ≠ 1. □

Our final preliminary result is a Mayer–Vietoris sequence:

Lemma 14.9. Let W be a complex G–representation with W^{G} = 0. Let W = W_{1} ⊕ W_{2} be a direct sum decomposition, where W_{1} = W_{max}.

(i) There is a long exact sequence

⋯ → L_{n+1}^{I}(C_{W ×R_{−},G}^{>W_{1}×R_{−}∪W_{2}×R_{−}}(Z))⟶^{∂_{n+1}}L_{n}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)) →

L_{n}^{s}(C_{W ×R_{−},G}^{>W_{1}×R_{−}}(Z)) ⊕ L_{n}^{s}(C_{W ×R_{−},G}^{>W_{2}×R_{−}}(Z)) → L_{n}^{I}(C_{W ×R_{−},G}^{>W′×R_{−}∪W′′×R_{−}}(Z))

of bounded L-groups, where

I = Im [K_{2}(C_{W ×R_{−},G}^{>W_{2}×R_{−}}(Z)) → K_{2}(C_{W ×R_{−},G}^{>W′×R_{−}∪W′′×R_{−}}(Z))]

is the decoration subgroup. (ii) For n ≡ 3 (mod 4), the boundary map ∂_{n+1} = 0.

Proof. The Mayer–Vietoris sequence in L^{⟨−∞⟩}

⋯ → L_{n+1}(C_{M,G}^{>U∪V} (Z)) → L_{n}(C_{M,G}^{>U∩V} (Z)) →

L_{n}(C_{M,G}^{>U}(Z)) ⊕ L_{n}(C_{M,G}^{>V}(Z)) → L_{n}(C_{M,G}^{>U∪V} (Z)

where U, V are nice G–invariant subspaces of the control space M, follows from an excision isomorphism

L_{n}(C_{U∪V,G}^{>U}(Z)) = L_{n}(C_{V,G}^{>U∩V} (Z))

and standard diagram-chasing. We apply this to M = W × R_{−}, U = W_{1} × R_{−} and V = W_{2} × R_{−}, where U ∩ V = R_{−}. The decorations follow from Section 6 and Corollary 8.5.

To see that ∂_{0} = 0, note that this boundary map is the composition of

∂: L_{0}^{I}(C_{W ×R_{−},G}^{>W′×R_{−}∪W′′×R_{−}}(Z)) → L_{3}^{s}(C_{W′×R_{−}∪W′′×R_{−},G}^{>W_{2}×R_{−}}(Z))

and an excision isomorphism

L_{3}^{s}(C_{W′×R_{−}∪W′′×R_{−},G}^{>W_{2}×R_{−}}(Z)) ≅ L_{3}^{s}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z)) .

But L_{3}^{s}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z)) = 0 by Lemma 14.7. □

Corollary 14.10. Suppose W is a complex G–representation with no R_{+} summands. Then the image of trf_{W ×R_{−}}(σ) is zero in L_{2k+1}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)), where dim W = 2k.

Proof. First recall from Theorem 3.6 that trf_{W ×R_{−}}(σ) ∈ L_{2k+1}^{c_{1}}(C_{W ×R_{−},G}(Z)) where c_{1} = Im(Wh(ZG) → Wh(C_{W ×R_{−},G}(Z))) under the cone point inclusion. Therefore, the image of trf_{W ×R_{−}}(σ) “away from R_{−}” lands in L_{2k+1}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)). We may therefore apply the Mayer–Vietoris sequence from Lemma 14.9.

As usual, we may assume that W has only 2–power isotropy subgroups, and we will argue by induction on dim W and on the orbit type filtration of W. The result is true for dim W = 2 (by Lemma 14.7) or for W a free representation (by a direct calculation following the method of Lemma 14.3). For sets of 2– power index subgroups (ordered by inclusion, with repetitions allowed), we say that {K′_{1}, …, K′_{s}} < {K_{1}, …, K_{t}} if s < t or s = t, K′_{i} = K_{i} for 1 ≦ i < j ≦ s, and K′_{j} ⫋ K_{j}. Let Iso(W) = {K_{1}, …, K_{t}} denote the set of isotropy subgroups of W ordered by inclusion. Define an ordering by W′ < W if (i) dim W′ < dim W or (ii) dim W′ = dim W but Iso(W′) < Iso(W).

Let W = W_{1} ⊕ W_{2}, where W_{1} has maximal isotropy subgroup K, and suppose the result holds for all W′ < W. If dim W ≡ 2 (mod 4), we get an injection

L_{3}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)) → L_{3}^{s}(C_{W ×R_{−},G}^{>W_{1}×R_{−}}(Z)) ⊕ L_{3}^{s}(C_{W ×R_{−},G}^{>W_{2}×R_{−}}(Z))

from Lemma 14.9, where W_{2} < W and W_{1} < W. If dim W ≡ 0 (mod 4), we get an injection

L_{3}^{s}(C_{W ×U×R_{−},G}^{>R_{−}}(Z)) → L_{3}^{s}(C_{W ×U×R_{−},G}^{>W_{1}×R_{−}}(Z)) ⊕ L_{3}^{s}(C_{W ×U×R_{−},G}^{>W_{2}×U×R_{−}}(Z))

again from Lemma 14.9, where U is any 2–dimensional free G–representation. Notice that W_{2} ⊕ U < W in the ordering above. Consider the following commutative diagram (with dim W ≡ 2 (mod 4)):

\xymatrix{L_{0}^{h}(ZG) \ar[r]^{^{trf^{W_{2}×R_{−}}}} \ar[dr]_{_{trf_{W ×R_{−}}}} & L_{1}^{c_{1}}(C_{W_{2}×R_{−},G}(Z)) \ar[r] \ar[d]^{_{trf_{W_{1}}}} & L_{1}^{s}(C_{W_{2}×R_{−},G}^{>R_{−}}(Z)) \ar[d]^{_{trf_{W_{1}}}} \\  & L_{3}^{c_{3}}(C_{W ×R_{−},G}(Z)) \ar[r] & L_{3}^{s}(C_{W ×R_{−},G}^{>W_{1}×R_{−}}(Z))}

This diagram, and our inductive assumption shows that the image of trf_{W ×R_{−}}(σ) is zero in L_{3}^{s}(C_{W ×R_{−},G}^{>W_{1}×R_{−}}(Z)). Similarly, by reversing the roles of W_{1} and W_{2} in the

diagram, we see that the image of trf_{W ×R_{−}}(σ) is zero in L_{3}^{s}(C_{W ×R_{−},G}^{>W_{2}×R_{−}}(Z)). There-

fore the image of trf_{W ×R_{−}}(σ) is zero in L_{3}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)). If dim W ≡ 0 (mod 4), we replace W by W ⊕ U and make a similar argument, using the observation that W_{2} ⊕ U < W to justify the inductive step. □

Proof of Theorem 14.1. Part (i) is an immediate consequence of Proposition 5.6, so we consider part (ii). From Corollary 14.10 we know that the image of our obstruction trf_{W ×R_{−}}(σ) is zero “away from R_{−}”, so comes from a torsion element in L_{2k+1}^{h}(C_{R_{−},G}(Z)) (by Lemma 14.3, L_{2k+2}(C_{W ×R_{−},G}^{>R_{−}}(Z)) is torsion). But L_{2k+1}^{h}(C_{R_{−},G}^{>0}(Z)) = L_{2k}^{p}(ZH) is torsion–free (except for the Arf invariant Z/2 which doesn’t matter), hence the image of trf_{W ×R_{−}}(σ) vanishes in L_{2k+1}^{h}(C_{W ×R_{−},G}^{>0}(Z)).

The exact sequence

⋯ → L_{2k+1}^{k}(ZG, w) → L_{2k+1}^{h}(C_{W ×R_{−},G}(Z)) → L_{2k+1}^{h}(C_{W ×R_{−},G}^{>0}(Z)),

with k = ker(K̃_{0}(ZG) → K̃_{0}(C_{W ×R_{−},G}(Z))), now shows that there exists a torsion element

σ̂ ∈ Im(L_{2k+1}^{k}(ZG, w) → L_{2k+1}^{p}(ZG, w))

with c_{∗}(σ̂) = trf_{W ×R_{−}}(σ) ∈ L_{2k+1}^{p}(C_{W ×R_{−},G}(Z)). □

Remark 14.11. In Theorem 16.7 we use the sharper result σ̂ ∈ L_{2k+1}^{k}(ZG, w) to improve our unstable similarity results by removing extra R_{−} factors. Since K_{1}(C_{R_{−},G}(Z)) → K_{1}(C_{W ×R_{−},G}(Z) is injective by Corollary 8.9, the decoration k = Im(K̃_{0}(ZH) → K̃_{0}(ZG)). □

15. The proof of Theorem C

By replacing V_{i} by V_{i} ⊕U if necesary, where U is free and 2–dimensional, we may assume that dim V_{i} ⊕W ≡ 0 ( mod 4). This uses the s–normal cobordism condition. By the top component argument (see Theorem 3.8) and Proposition 9.7, we may also assume that W has only 2–power isotropy. We have a commutative diagram analogous to (13.1).

\xymatrix{ & H^{1}(K̃_{0}(C_{W ×R_{−},G}(Z))) \ar[rr]^{δ^{2}} \ar@{-->}[dr] &  & H^{1}(K_{1}(C_{W ×R_{−},G}(Z)) \ar[rr] \ar[dr] &  & L_{0}^{s}(C_{W ×R_{−},G}(Z)) \\ (15.1) &  & L_{1}^{h}(C_{W ×R_{−},G}(Z)) \ar@{-->}[ur] \ar[dr] &  & H^{1}(Δ_{W ×R_{−}}) \ar@{-->}[ur] \ar[dr] \\  & L_{1}^{s}(C_{W ×R_{−},G}(Z)) \ar[ur] \ar[rr] &  & L_{1}^{p}(C_{W ×R_{−},G}(Z)) \ar[ur] \ar[rr] &  & H^{0}(K̃_{0}(C_{W ×R_{−},G}(Z)))}

where H^{1}(Δ_{W ×R_{−}}) denotes the relative group of the double coboundary map.

This diagram for W can be compared with the one for W_{1} = W_{max} via the inclusion maps, and we see that the K-theory terms map isomorphically by Lemma 8.2. Now by Theorem 14.1 there exists a torsion element

σ̂ ∈ Im(L_{1}^{p}(C_{R_{−},G}(Z)) → L_{1}^{p}(C_{W_{1}×R_{−},G}(Z)))

which hits trf_{W ×R_{−}} under the cone point inclusion. In particular, σ̂ vanishes “away from R_{−}”.

We have the exact sequence

L_{2}^{s}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z)) → L_{1}^{Y} (C_{R_{−},G}(Z)) → L_{1}^{s}(C_{W_{1}×R_{−},G}(Z)) → L_{1}^{s}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z))

and Y = ker(Wh(C_{R_{−},G}(Z)) → Wh(C_{W_{1}×R_{−},G}(Z)) is zero, by Corollary 8.9. But L_{2}^{s}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z)) is a torsion group, by Lemma 14.3, and L_{1}^{Y} (C_{R_{−},G}(Z)) = L_{1}^{s}(C_{R_{−},G}(Z)) is torsion–free by Lemma 12.1 so L_{1}^{Y} (C_{R_{−},G}(Z)) ↣ L_{1}^{s}(C_{W_{1}×R_{−},G}(Z)) is an injection.

Since L_{1}^{s}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z)) = Z/2 by Remark 14.8 we conclude that the previous group L_{1}^{s}(C_{W_{1}×R_{−},G}(Z)) is torsion–free modulo this Z/2, which injects into L1^{⟨−1⟩}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z)) = Z/2 ⊕ H^{1}(K_{−1}(ZK)). Since σ̂ vanishes away from R_{−}, this extra Z/2 may be ignored.

By substituting this computation into the exact sequence

L_{1}^{s}(C_{W_{1}×R_{−},G}(Z)) → L_{1}^{p}(C_{W_{1}×R_{−},G}(Z)) → H^{1}(Δ_{W_{1}×R_{−}})

of (13.1), we see that the torsion subgroup of

ker(L_{1}^{p}(C_{W_{1}×R_{−},G}(Z)) → L_{1}^{p}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z)))

injects into the relative group H^{1}(Δ_{W_{1}×R_{−}}) of the double coboundary. Therefore σ̂ ∈ L_{1}^{p}(C_{W_{1}×R_{−},G}(Z)) vanishes if and only if the element {Δ(V_{1})/Δ(V_{2})} is in the image of the double coboundary

δ^{2}: H^{1}(K̃_{0}(C_{W_{max}×R_{−},G}(Z))) → H^{1}(Wh(C_{W_{max}×R_{−},G}(Z))) □

A very similar argument can be used to give an inductive criterion for non–linear similarity without the R_{+} summand (generalizing Theorem A). In the statement we will use the analogue to k, namely k_{W} = ker(Wh(C_{W ×R_{−},G}^{>0}(Z)) → K̃_{0}(ZG)). It follows from our K–theory calculations that k_{W} = k when K_{−1}(ZK) = 0 for K ∈ Iso(W).

Theorem 15.2. Let V_{1} = t^{a_{1}} + ⋯ + t^{a_{k}} and V_{2} = t^{b_{1}} + ⋯ + t^{b_{k}} be free G– representations. Let W be a complex G–representation with no R_{+} summands. Then there exists a topological similarity V_{1} ⊕ W ⊕ R_{−} ∼_{t} V_{2} ⊕ W ⊕ R_{−} if and only if

(i) S(V_{1}) is s–normally cobordant to S(V_{2}), (ii) Res_{H}(V_{1} ⊕ W) ⊕ R_{+} ∼_{t} Res_{H}(V_{2} ⊕ W) ⊕ R_{+}, and (iii) the element {Δ(V_{1})/Δ(V_{2})} is in the image of the coboundary

δ: H^{0}(k_{W_{max}}) → H^{1}(Wh(C_{W_{max}×R_{−},G}(Z))) ,

where 0 ⊆ W_{max} ⊆ W is a 2–dimensional complex subrepresentation with maximal isotropy group among the isotropy groups of W with 2-power index.

Proof. We leave the details to the reader. □

16. Iterated R_{−} transfers

We now apply Theorem 7.10 to show that iterated R_{−} transfers don’t lead to any new similarities.

Theorem 16.1. Suppose that W is a complex G–representation with no R_{+} summands. Then V_{1} ⊕ W ⊕ R_{−}^{l} ⊕ R_{+} ∼_{t} V_{2} ⊕ W ⊕ R_{−}^{l} ⊕ R_{+} for l ≧ 1 implies V_{1} ⊕ W ⊕ R_{−} ⊕ R_{+} ∼_{t} V_{2} ⊕ W ⊕ R_{−} ⊕ R_{+}.

The first step in the proof is to show injectivity of certain transfer maps. For any homomorphism w: G → {±1}, we will use the notation (C_{W,G}(Z), w) to denote the antistructure where the involution is g ↦ w(g)g^{−1} at the cone point. The standard orientation (6.4) has w = det(ρ_{W}), but we will need others in this section. Let ϕ = det(ρ_{R_{−}}) for short, and notice that ϕ: G → {±1} is non–trivial.

Lemma 16.2. The transfer map trf_{R_{−}}: L_{2k+1}^{p}(ZG, w) → L_{2k+2}^{p}(C_{R_{−},G}(Z), wϕ) is injective, where w is the non–trivial orientation.

Proof. Since L_{2k+2}^{p}(C_{R_{−},G}(Z), w) ≅ L_{2k+2}^{p}(ZH → ZG), we just have to check that the previous map in (7.6) is zero (using the tables in [20]). □

Next, a similar result for the R_{−}^{2} transfer.

Proposition 16.3. The bounded transfer

trf_{R_{−}^{2}}: L_{2k+1}^{p}(ZG, w) → L_{2k+3}^{p}(C_{R_{−}^{2},G}(Z), w)

is injective, where w is the non–trivial orientation.

Proof. We can relate the iterated R_{−} transfer trf_{1} ∘ trf_{1} to trf_{2} = trf_{R_{−}^{2}} by means of the braid diagram:

\xymatrix{ & L_{n}(ZG, w) \ar[rr]^{^{trf^{2}}} \ar[dr]^{_{trf_{1}}} &  & L_{n+2}(ZΓ_{H} →ZG, w) \ar[rr] \ar[dr] &  & LNS_{n+3}(Φ) \\ (16.4) &  & L_{n+1}(ZH →ZG, wϕ) \ar[ur]^{_{trf_{1}}} \ar[dr] &  & LS_{n+3}(Φ) \ar[ur] \ar[dr] \\  & LNS_{n}(Φ) \ar[ur] \ar[rr] &  & LN_{n+3}(ZH →ZG, wϕ) \ar[ur] \ar[rr] &  & L_{n+3}(ZG, w)}

where the new groups LNS_{∗}(Φ) are the relative groups of the transfer

trf_{1} = trf_{R_{−}}: L_{k}^{p}(C_{R_{−},G}(Z), wϕ) → L_{k+1}^{p}(C_{R_{−}^{2},G}(Z), w) .

The diagram Φ of groups (as in [38,Chap.11]) contains Γ_{H} ≅ Z × H as the fundamental group of S(R_{−} ⊕ R_{−})/G (see [19,5.9]).

The groups LNS_{∗}(Φ) have a geometric bordism description in terms of triples L^{n−2} ⊂ N^{n−1} ⊂ M^{n} where each manifold has fundamental group G, N is characteristic in M, and L is characteristic in N, with respect to the index two inclusion H < G. There is also an algebraic description for LNS_{∗}(Φ) in terms of “twisted antistructures” [22] as for the other groups LN(ZH → ZG, wϕ) [38,12C] and for LS(Φ, w) [27,7.8.12]. Substituting these descriptions into the braid gives:

\xymatrix{ & L_{n}(ZG, α, u) \ar[rr]^{^{trf^{2}}} \ar[dr]^{_{trf_{1}}} &  & L_{n}(ZΓ_{H} →ZG, α̃, ũ) \ar[rr] \ar[dr] &  & L_{n−1}(ZH →ZΓ_{H}, α̃, ũ) \\ (16.5) &  & L_{n}(ZH →ZG, α, u) \ar[ur]^{_{trf_{1}}} \ar[dr] &  & L_{n−1}(ZΓ_{H}, α̃, ũ) \ar[ur] \ar[dr] \\  & L_{n}(ZH →ZΓ_{H}, α̃, ũ) \ar[ur] \ar[rr] &  & L_{n−1}(ZH, α, u) \ar[ur] \ar[rr] &  & L_{n−1}(ZG, α, u)}

The antistructure (ZG, α, u) is the twisted antistructure obtained by scaling with an element a ∈ G − H. The antistructure ZΓ_{H}, α̃, ũ) is the one defined by Ranicki [27,p.805], then scaled by ã ∈ Γ_{H}, where ã maps to a under the projection Γ_{H} → G. Since Γ_{H} ≅ Z × H, we have an exact sequence

→ L_{n}^{p}(ZH, α, u) → L_{n}^{p}(ZΓ_{H}, α̃, ũ) → Ln−1^{⟨−1⟩}(ZH, α, u)⟶^{1−w(a)}L_{n−1}^{p}(ZH, α, u)

and it follows that L_{n}^{p}(ZH →ZΓ_{H}, α̃, ũ) ≅ Ln−1^{⟨−1⟩}(ZH, α, u). It is not difficult to see that L2k^{⟨−1⟩}(ZH, α, u) is torsion–free using the tables in [20].

Now observe that L_{n}^{p}(ZG, α, u) ≅ L_{n}^{p}(ZG, w), and

L_{1}^{p}(ZH →ZG, α, u) ≅ L_{2}^{p}(ZH →ZG) .

The transfer trf_{R_{−}}: L_{1}^{p}(ZG, w) → L_{2}^{p}(ZH → ZG) is injective by Lemma 16.2, and since w(a) = −1, the map

L0^{⟨−1⟩}(ZH, α, u)⟶^{1−w(a)}L_{0}^{p}(ZH, α, u)

is also injective. Since L_{1}^{p}(ZG, w) is 2–torsion, it must inject into L_{1}^{p}(ZΓ_{H} → ZG, α̃, ũ) ≅ L_{3}^{p}(C_{R_{−}^{2},G}(Z), w). □

The final step in the proof is to consider the following commutative square of spectra:

\xymatrix{𝕃^{p}(C_{pt}(ZG), w) \ar[r]^{c_{∗}} \ar[d]_{_{trf_{R−}}} & 𝕃^{p}(C_{W ×R_{−},G}(Z), w) \ar[d]^{_{trf_{R−}}} \\ 𝕃^{p}(C_{R_{−},G}(Z), wϕ) \ar[r]^{c_{∗}} & 𝕃^{p}(C_{W ×R_{−}^{2},G}(Z), wϕ)}

for l ≥ 1.

Lemma 16.6. This is a pull–back square.

Proof. The fibres of the vertical R_{−} transfers are

𝕃ℕ^{p}(C_{R_{−},G}(Z), wϕ) ≅ 𝕃^{p}(C_{pt}(ZH, α, u)) ,

and 𝕃ℕ^{p}(C_{W ×R_{−},G}(Z), wϕ) ≅ 𝕃^{p}(C_{W,H}(Z), α, u) respectively by (7.6). The fibre of the cone point inclusion

𝕃^{p}(C_{pt}(ZH), α, u) → 𝕃^{p}(C_{W,H}(Z), α, u)

is 𝕃^{h}(C_{W,H}^{>0}(Z), α, u), since K_{0}(C_{W,H}^{>0}(Z)) → K_{−1}(C_{pt}(ZH) is injective. But Theorem 7.10 shows that 𝕃^{h}(C_{W,H}^{>0}(Z), α, u) is contractible, and therefore the cone point inclusion is a homotopy equivalence. It follows that the fibres of the horizontal maps are also homotopy equivalent. □

Proof of Theorem 16.1. It is enough to prove that trf_{W ×R_{−}^{l+1}}(σ) = 0. for l ≧ 1, implies trf_{W ×R_{−}}(σ) = 0 in the top component of L_{2k+1}^{p}(C_{W ×R_{−},G}(Z)). We may assume that Iso(W) only contains subgroups of 2–power index, and let dim W = 2k. Now by Theorem C, trf_{W ×R_{−}^{2t+1}}(σ) = 0 implies trf_{W ×R_{−}^{2t−1}}(σ) = 0, provided that t ≧ 2. It therefore remains to study l = 1 and l = 2.

We have the commutative diagram:

\xymatrix{L_{2k+1}^{p}(ZG, w) \ar[r]^{c_{∗}} \ar[d]_{_{trf_{Rl−}}} & L_{2k+1}^{p}(C_{W ×R_{−},G}(Z), w) \ar[d]^{_{trf_{Rl−}}} \\ L_{2k+l+1}^{p}(C_{R_{−}^{l},G}(Z), wϕ^{l}) \ar[r]^{c_{∗}} & L_{2k+l+1}^{p}(C_{W ×R_{−}^{l+1},G}(Z), wϕ^{l})}

In the lower row, the orientation at the cone point is non–trivial for l even, and trivial for l odd. Suppose first that l = 1. Then, the transfer map

trf_{R_{−}}: L_{2k+1}^{p}(ZG, w) → L_{2k+2}^{p}(C_{R_{−},G}(Z), wϕ) = L_{2k+2}^{p}(ZH →ZG)

is injective by Lemma 16.2. Since trf_{W ×R_{−}^{l}}(σ) = c_{∗}(σ̂) for some σ̂ ∈ L_{2k+1}^{p}(ZG, w), we are done by a diagram chase (using Lemma 16.6 to see that the horizontal maps sit in a commutative ladder with isomorphisms every third term).

If l = 2, we use the fact that trf_{R_{−}^{2}}: L_{2k+1}^{p}(ZG, w) → L_{2k+3}^{p}(C_{R_{−}^{2},G}(Z), w) is injective, from Proposition 16.3. A similar diagram chase completes the proof. □

We also have a version without R_{+} summands.

Theorem 16.7. Suppose that W is a complex G–representation with W^{G} = 0. Then V_{1} ⊕W ⊕R_{−}^{l} ∼_{t} V_{2} ⊕W ⊕R_{−}^{l} for l ≧ 1 implies V_{1} ⊕W ⊕R_{−} ∼_{t} V_{2} ⊕W ⊕R_{−}.

Proof. This follows from a similar argument, using Theorem 15.2 instead of Theorem C. The injectivity results of Lemma 16.2 and Proposition 16.3 also hold for trf_{1} and trf_{2} on L_{2k+1}^{k}(ZG, w), where k = Im(K̃_{0}(ZH) → K̃_{0}(ZG)) as in Remark 14.11. The details are left to the reader. □

17. The proof of Corollary 2.4.

Suppose that G = C(4q), q odd, and that V_{1} ⊕ R_{−} ⊕ R_{+} ∼_{t} V_{2} ⊕ R_{−} ⊕ R_{+} with dim V_{i} = 4. We will use Theorem B and the assumption about odd class numbers to proved that V_{1} ≅ V_{2}. This is the only case we need to discuss to prove Corollary 2.4. If we started instead with V_{1} ⊕ R_{−}^{2} ∼_{t} V_{2} ⊕ R_{−}^{2}, then stabilizing with R_{+} and applying Theorem 16.1 would reduce to the case above.

We may assume that q > 1 is a prime, and that the free representations have the form V_{1} = t + t^{i} and V_{2} = t^{1+2q} + t^{i+2q} where (i, 2q) = 1. The Reidemeister torsion invariant is Δ(V_{1})/Δ(V_{2}) = U_{1,i} ∈ Wh(ZG) in the notation for units of ZG used in [7,p.732]. With respect to the involution : t ↦ −t^{−1}, this element defines a class u_{1,i} = {U_{1,i}} ∈ H^{1}(Wh(ZG^{−})). Since the map L_{1}^{p}(ZG^{−}) → L_{1}^{p}(C_{R_{−},G}(Z)) is injective, it is enough to show that the image of u_{1,i} is non–trivial in H^{1}(Δ) (see diagram (11.6)).

Let A = Z[ζ_{4q}] be the ring of cyclotomic integers., and B = Z[ζ_{2q}]. We will study the top component Wh(ZG)(q) by comparing it to K_{1}(M), where M = A × B × B is the top component of an involution invariant maximal order in QG containing ZG. Notice that the two copies of B are interchanged under the involution, so H^{i}(K_{1}(M)) = H^{i}(A^{×}). We have the exact sequence

0 → Wh(ZG)(q) → K_{1}(M) → K_{1}(M̂)/ Wh(ẐG)(q) → D(ZG)(q) → 0 .

where D(ZG) = ker(K̃_{0}(ZG) → K̃_{0}(M)). Note that H^{i}(D(ZG)) = H^{i}(K̃_{0}(ZG)) since A has odd class number.

Let i_{∗}: H^{1}(Wh(ZG^{−})(q)) → H^{1}(A^{×}) denote the map on the Tate cohomology induced by the inclusion i: Wh(ZG)(q) → A^{×}.

Lemma 17.1. For (j, 2q) = 1 and j ≡ 1 (mod 4), the image i_{∗}(u_{1,j}) = ⟨−1⟩ ∈ H^{1}(A^{×}).

Proof. Let γ_{j} = \frac{t^{j}−1}{t−1} ∈ A^{×}, and compute γ̄_{j}/γ_{j} with respect to the non–oriented involution. We get γ̄_{j}/γ_{j} = u_{1,j}/u_{1,1}, so i_{∗}(u_{1,j}) = i_{∗}(u_{1,1}) for all i. Now let v = \frac{t+1}{t−1}. Since t = −t^{1+2q}, this is a cyclotomic unit in A with vv̄ = −1. But v/v̄ = −u_{1,1}, so i_{∗}(u_{1,j}) = ⟨−1⟩ in H^{1}(A^{×}). □

Next we need a computation:

Lemma 17.2. 0 ≠ ⟨−1⟩ ∈ H^{1}(A^{×}).

Proof. Let E = Q(ζ_{4q}) be the quotient field of A. Since A has odd class number, H^{i}(E^{×}/A^{×}) ≅ H^{i}(Ê^{×}/Â^{×}). Next, observe that the extension E/F is evenly ramified at q, where F is the fixed field of the involution : E → E. It follows that the map H^{0}(Êq^{×}) → H^{0}(Êq^{×}/Âq^{×}) is zero. We finish by considering the commutative diagram

\xymatrix{H^{0}(E^{×}) \ar[r] \ar[d] & H^{0}(E^{×}/A^{×}) \ar[r]^{δ} \ar[d]^{≈} & H^{1}(A^{×}) \\ H^{0}(Ê^{×}) \ar[r] & H^{0}(Ê^{×}/Â^{×})}

and the prime element w = ζ_{q} − ζ_{q}^{−1} ∈ E over q. Since w̄/w = −1 it defines an element in H^{0}(E^{×}/A^{×}) and δ(w) = ⟨−1⟩ ∈ H^{1}(A^{×}). On the other hand, w maps under the middle isomorphism to a non–zero element in H^{0}(Êq^{×}/Âq^{×}) so δ(w) ≠ 0 □

Now we can complete the proof of Corollary 2.4 by considering the commutative braid:

\xymatrix{ & H^{0}(A^{×}) \ar[rr] \ar[dr] &  & H^{0} (\frac{K_{1}(M̂)}{Wh(ẐG)(q)}) \ar[rr] \ar@{-->}[dr] &  & H^{0}(K̃_{0}(ZG)(q)) \\ (17.3) &  & H^{0} (\frac{K_{1}(M)}{Wh(ZG)(q)}) \ar@{-->}[ur] \ar@{-->}[dr] &  & H^{1}(Δ) \ar[ur] \ar[dr] \\  & H^{1}(K̃_{0}(ZG)(q)) \ar[ur] \ar[rr]_{δ^{2}} &  & H^{1}(Wh(ZG^{−})(q)) \ar[ur] \ar[rr]_{i_{∗}} &  & H^{1}(A^{×})}

containing the double coboundary and its relative group H^{1}(Δ). By commutativity of the lower right triangle, the image of our obstruction element u_{1,j} is non–zero in H^{1}(A^{×}) and therefore non–zero in H^{1}(Δ). □

18. The normal invariant.

In this section we collect some results about normal cobordisms of lens spaces with cyclic fundamental group G. In [6,§1] it is asserted that their formula (A′) gives necessary and sufficient conditions for homotopy equivalent lens spaces to be s–normally cobordant, when G = C(2^{r}). However in [40] it is shown that the given conditions (A′) are sufficient but not necessary. We only use the sufficiency here.

We want to study the subgroups R̃^{free}(G) and R̃h^{free}(G) of R(G).

Lemma 18.1. Let G = C(2^{r}q) be a finite cyclic group, and G_{2} be the 2–Sylow subgroup. If q > 1 there is an exact sequence

0 → Z/2 → R̃^{free}(G)/R̃h^{free}(G) → R̃^{free}(G_{2})/R̃h^{free}(G_{2}) → 0

given by the restriction map Res_{G_{2}}. The kernel is generated by any element α ∈ R̃^{free}(G) with k–invariant k(α) ∈ (Z/|G|)^{×}/{±1} in the coset k ≡ 1 (mod 2^{r}) and k ≡ −1 (mod q).

Proof. There is a short exact sequence

0 → Rh^{free}(G) → R̃^{free}(G) → (cy2^{r}q)^{×}/{±1}

given by the k–invariant. Moreover, the k–invariants of elements in R̃^{free}(G) lie in

ker((Z/2^{r}q)^{×}/{±1} → (Z/q)^{×}/{±1}) ≅ {lq + 1 | l ∈ Z} ⊆ (Z/2^{r}q)^{×}

which injects under reduction (mod 2^{r}) into (Z/2^{r})^{×} if q > 1. Restriction of the k–invariant to G_{2} detects only its value in (Z/2^{r})^{×}/{±1}, so the kernel has order 2. Since the restriction map R̃^{free}(G) → R̃^{free}(G_{2}) is surjective, we have the required exact sequence. □

The situation for the normal invariant is simpler. Recall that we write V ≏ V′ or (V − V′) ≏ 0 if there exists a homotopy equivalence f: S(V)/G → S(V′)/G of lens spaces, such that f is s–normally cobordant to the identity.

Lemma 18.2. Let G = C(2^{r}q) be a finite cyclic group. Then

ker(Rh^{free}(G) → Rh^{free}(C(q)) ⊕ Rh^{free}(C(2^{r}))) ⊆ Rn^{free}(G) .

Proof. We may assume that r ≥ 2, since Res: R^{free}(C(2q)) → R^{free}(C(q)) is an isomorphism, and consider an element

∑ t^{a_{i}} − ∑ t^{b_{i}} ∈ ker(R^{free}(G) → R^{free}(C(q)) ⊕ R^{free}(C(2^{r}))).

With a suitable ordering of the indices, a_{i} ≡ b_{i} (mod 2^{r}) and a_{i} ≡ b_{τ(i)} (mod q) for some permutation τ. Since the normal invariants of lens spaces are detected by a cohomology theory, we can check the condition at each Sylow subgroup separately. It follows that ∑ t^{a_{i}} ≏ ∑ t^{b_{i}}. □

Example 18.3. For G = C(24) we have R̃^{free}(G) = {t − t^{5}, t − t^{7}, t − t^{11}} and ker Res_{C(8)} ∩R̃^{free}(G) = {t − t^{7}, t^{5} − t^{11}}. □

Lemma 18.4. Let G = C(2^{r}q) be a finite cyclic group, and G_{2} be the 2–Sylow subgroup. If r ≥ 2 there is an isomorphism

R̃h^{free}(G)/R̃n^{free}(G)) → R̃h^{free}(G_{2})/R̃n^{free}(G_{2})

given by the restriction map Res_{G_{2}}.

Proof. We first remark that if W = t^{a_{1}} + ⋯ + t^{a_{n}} is a free G_{2}–representation (here we use the assumption that r ≥ 2), then by choosing integers b_{i} ≡ a_{i} (mod 2^{r}) and b_{i} ≡ 1 (mod q) we obtain a free G–representation V = t^{b_{1}} + ⋯ + t^{b_{n}} with Res_{G_{2}}(V) = W. It follows that

Res_{G_{2}}: R̃h^{free}(G)/R̃n^{free}(G)) → R̃h^{free}(G_{2})/R̃n^{free}(G_{2})

is surjective.

Now suppose α ∈ R̃h^{free}(G) and Res_{G_{2}}(α) ∈ Rn^{free}(G_{2}). This means that

Res_{G_{2}}(α) = ∑ β_{i}

where each β_{i} = W_{i} − W′_{i} for some free G_{2}–representations W_{i}, W′_{i} such that W_{i} ≏ W′_{i}. By the construction of the last paragraph, we can find free G–representations V_{i}, V′_{i} such that (i) Res_{G_{2}}(V_{i}) = W_{i} and Res_{G_{2}}(V′_{i}) = W′_{i}, and (ii) (V_{i} − V′_{i}) ∈ R̃h^{free}(G). It follows that V_{i} ≏ V′_{i} and so α′ = ∑ α_{i} ∈ R̃n^{free}(G)), and

α − α′ ∈ ker(Rh^{free}(G) → Rh^{free}(C(q)) ⊕ Rh^{free}(C(2^{r}))).

By Lemma 18.2, α − α′ ∈ Rn^{free}(G). □

Our final result is a step towards determining R_{h}(G)/R_{n}(G) more explicitly.

Lemma 18.5. Let b_{i,s} = (t^{i} − t^{2^{r−s}q−i}) ∈ R̃^{free}(G) for 1 ≤ i < 2^{r−s−1}q, 1 ≤ s ≤ r − 1, and (i, 2q) = 1. Let l(i, s) = s for 1 ≤ s ≤ r − 2, and l(i, r − 1) the order of i/(2q − i) ∈ (Z/2^{r})^{×}. Then b_{i,s} is an element of order 2^{l(i,s)} in R(G)/R_{h}(G), and 2^{l(i,s)} · b_{i,s} ≏ 0.

Proof. If 1 ≤ s ≤ r − 2 then k(b_{i,s}) = i/(2^{r−s}q − i) ∈ (Z/2^{r}q)^{×}/{±1} is congruent to −1 (mod 4q). By Lemma 18.1 the linear span of these elements in R(G)/R_{h}(G) injects into R(G_{2})/R_{h}(G_{2}), where G_{2} is the 2–Sylow subgroup. By Lemma 18.4 the normal invariant is also detected by restriction to G_{2}, so it is enough to prove the assertions about these elements (s ≤ r − 2) when G = C(2^{r}).

For the first part we must show that the k–invariant of b_{i,s} has order 2^{s}. We will use the expression

ν_{2}(r!) = r − α_{2}(r),

where α_{2}(r) is the number of non-zero coefficients in the 2-adic expansion of r, for the 2-adic valuation of r. Then

ν_{2}(\binom{2^{s}}{k}) = s − ν_{2}(k)

for 1 ≤ k ≤ 2^{s}. These formulas and the binomial expansion show that k(b_{i,s})^{2^{s}} ≡ 1 (mod 2^{r}) for 1 ≤ s ≤ r − 2.

Next we consider the normal invariant. For 1 ≤ s ≤ r − 2 we will take q = 1 and apply the criterion (A′) of [6] to show that 2^{s} · b_{i,s} ≈ 0. This amounts to a re-labelling of our original elements b_{i,s} without changing the order of their k– invariants.

We must now compute the elementary symmetric functions σ_{k}(2^{s} ·i^{2}) and σ_{k}(2^{s} · (2^{r−s} −i)^{2}), where the notation 2^{s} ·i^{2} means that the weight i^{2} is repeated 2^{s} times in the symmetric function. The formula (A′) is

σ_{k}(2^{s} · (2^{r−s} + i)^{2}) − σ_{k}(2^{s} · i^{2}) ≡ 2((2^{r−s} − i)^{2^{s}} − i^{2^{s}})\binom{2^{s} − 1}{k − 1} (mod 2^{r+3})

so on the right-hand side we have

\{ \begin{matrix}+2^{r+1} (mod 2^{r+3}) & if k odd \\ −2^{r+1} (mod 2^{r+3}) & if k even \end{matrix}

The formula (A′) assumes that the weights are congruent to 1 (mod 4), or in our case i ≡ 1 ( mod 4). Therefore, if i ≡ 3 ( mod 4), we must use the equivalent weights −i and −(2^{r−s} − i).

To compute the left-hand side we use the Newton polynomials s_{k} and their expressions in term of elementary symmetric functions. We need the property s_{k}(2^{s} · i^{2}) = 2^{s}i^{2k} and the coefficient of σ_{k} in s_{k} which is (−1)^{k+1}k. By induction, we see that the left-hand side is just

\frac{(−1)^{k+1}}{k} (s_{k}(2^{s}·(2^{r−s}−i)^{2})−s_{k}(2^{s}·i^{2})) ≡ \frac{(−1)^{k+1}2^{s}}{k} ((2^{r−s}−i)^{2k}−i^{2k}) ( mod 2^{r+3})

But by writing

(2^{r−s} − i)^{2k} − i^{2k} = (2^{r−s}θ + i^{2})^{k} − i^{2k}

where θ = 2^{r−s} − 2i ≡ 2 (mod 4), our expression becomes

\frac{(−1)^{k+1}2^{s}}{k} (2^{r−s}kθ) ≡ (−1)^{k+1}2^{r+1} (mod 2^{r+3})

If s = r − 1, then k(b_{i,s}) = i/(2q − i) ≡ −1 (mod q), and k(b_{i,s}) ≡ 1 (mod 2^{r}) whenever i ≡ q (mod 2^{r−1}). This is for example always the case for G = C(4q). Such elements b_{i,r−1} lie in the kernel of the restriction map to R(C(2^{r})). Moreover, if b_{i,r−1} ∈ ker(R(G) → R(C(2^{r}))), then 2b_{i,r−1} ∈ R_{h}(G) and 2b_{i,r−1} ≏ 0 by Lemma 18.2. Otherwise, the order of b_{i,r−1} ∈ R(G)/R_{h}(G) is the same as the order of its restriction Res(b_{i,r−1}) to the 2–Sylow subgroup, and Res(b_{i,r−1}) = ± Res(b_{j,s}) for l(i, r − 1) = s ≤ r − 2 and some j. □

19. Cyclic 2–groups.

For G = C(2^{r}) a cyclic 2–group, we have stronger results because K_{−1}(ZG) = Theorem 19.1. Let V_{1} = t^{a_{1}} + ⋯ + t^{a_{k}} and V_{2} = t^{b_{1}} + ⋯ + t^{b_{k}} be free G– representations, where G = C(2^{r}). Let W be a complex G–representation with no R_{+} or R_{−} summands. If Res_{H} V_{1} ≅ Res_{H} V_{2} then V_{1} ⊕ W ⊕ R_{−} ∼_{t} V_{2} ⊕ W ⊕ R_{−} implies V_{1} ≅ V_{2}.

Proof. We will show that the last criterion of Theorem 15.2 can not be fulfilled unless V_{1} ≅ V_{2}. First, since G = C(2^{r}) the obstruction map δ in Theorem 15.2 is just the coboundary δ: H^{0}(k) → H^{1}(Wh(ZG^{−})/ Wh(ZH)) used in Theorem A. Since Res_{H} V_{1} ≅ Res_{H} V_{2}, our Reidemeister torsion element {Δ(V_{1})/Δ(V_{2})} ∈ H^{1}(Wh(ZG^{−})), and therefore it vanishes under the composite

H^{1}(Wh(ZG^{−})) → H^{1}(Wh(ZG^{−})/ Wh(ZH)) → H^{0}(Wh(ZH)) .

Next, we claim that the composite

H^{0}(k)⟶^{δ}H^{1}(Wh(ZG^{−})/ Wh(ZH)) → H^{0}(Wh(ZH))

is injective. This map is the double coboundary associated to the sequence

0 → Wh(ZH) → Wh(ZG^{−}) → Wh(ZH →ZG^{−}) → k → 0

which by Section 9 equals the composite

H^{0}(k) → L_{0}^{h}(ZH) → H^{0}(Wh(ZH))

of maps in Ranicki–Rothenberg sequences. But the group L_{1}^{k}(ZH) = 0 by Lemma 11.10, so H^{0}(k) injects into the torsion subgroup of L_{0}^{h}(ZH), which is mapped injectively into H^{0}(Wh(ZH)) since L_{0}^{s}(ZH) is torsion–free. Therefore {Δ(V_{1})/Δ(V_{2})} is not in the image of δ and no such non–linear similarity exists. □

Corollary 19.2. The groups G = C(2^{r}) have no 5–dimensional non–linear similarities.

The 6–dimensional case was worked out in detail for cyclic 2–groups in [7], and a solution for the general case was announced in [6]. However, in Example 19.6 we give a counterexample to the necessity of [6,Condition B], and this invaliates the claimed solution.

Theorem 19.3. Let V_{1} = t^{a_{1}} + ⋯ + t^{a_{k}} and V_{2} = t^{b_{1}} + ⋯ + t^{b_{k}} be free G– representations, where G = C(2^{r}). Let W be a complex G–representation with no R_{+} summands. Then there exists a topological similarity V_{1} ⊕ W ⊕ R_{−} ⊕ R_{+} ∼_{t} V_{2} ⊕ W ⊕ R_{−} ⊕ R_{+} if and only if

(i) S(V_{1}) is s–normally cobordant to S(V_{2}), (ii) Res_{H}(V_{1} ⊕ W) ⊕ R_{+} ∼_{t} Res_{H}(V_{2} ⊕ W) ⊕ R_{+}, and (iii) the element {Δ(V_{1})/Δ(V_{2})} is in the image of the double coboundary

δ^{2}: H^{1}(K̃_{0}(ZH →ZG^{−})) → H^{1}(Wh(ZG^{−})/ Wh(ZH)) .

Corollary 19.4. There is a stable topological similarity V_{1} ≈_{t} V_{2} if and only if S(V_{1}) is s–normally cobordant to S(V_{2}) and Res_{K} {Δ(V_{1})/Δ(V_{2})} is in the image of the double coboundary δ_{K}^{2} for all subgroups K ⊆ G.

Remark 19.5. The double coboundary δ_{K}^{2} in this statement is the one for the subgroup K with respect to an index two subgroup K_{1} < K. Corollary 19.4 follows from Theorem 19.3 and Proposition 9.6. Notice that Theorem 19.3 also gives a way to construct the stable similarity, assuming that the conditions are satisfied. In the most complicated case, one would need all proper subgroups of G appearing as isotropy groups in W. □

The proof of Theorem 19.3. By Corollary 8.9 the double coboundary

δ_{W}^{2} : H^{1}(K̃_{0}(C_{W_{max}×R_{−},G}(Z))) → H^{1}(Wh(C_{W_{max}×R_{−},G}(Z)))

is isomorphic to

δ^{2}: H^{1}(K̃_{0}(ZH →ZG^{−})) → H^{1}(Wh(ZG^{−})/ Wh(ZH))

under the cone point inclusion. The result now follows from Theorem C. □

We conclude with two examples.

Example 19.6. For r ≥ 5 the element (t^{9} + t^{1+2^{r−2}} − t − t^{9+2^{r−2}}) lies in R̃t^{free}(G). On the other hand, the Reidemeister torsion invariant restricted to Wh(ZH) is given by U_{9,1+2^{r−2}} = U_{1,9}(U_{1,1})^{−1} and this is non–trivial in H^{0}(Wh(ZH)) by the results of Cappell–Shaneson on units (see [7,p.733]).

To prove the first assertion, note that (t^{9}+t^{1+2^{r−2}}−t−t^{9+2^{r−2}}) lies in R̃n^{free}(G) by applying [6,Condition A]. Let W denote the complex 2–dimensional representation with isotropy of index 4. The surgery obstruction in L_{3}^{p}(C_{W ×R_{−},G}(Z)) is zero by a similar argument to that given in [7,p.734], based on the fact that 9(1 + 2^{r−2}) ≡ 1 ( mod 8) when r ≥ 5 and invariance under group automorphisms (similar to [7,4.1] since our surgery obstruction comes from the cone point by Theorem 14.1). □

Example 19.7. According to [6,Thm.2], the element t − t^{5} has order 2^{r−2} in R_{Top}(G) for G = C(2^{r}), r > 3, and order 4 for G = C(8). Note that 2^{r−2}(t − t^{5}) ∈ R̃h^{free}(G), for r ≧ 3 and this is the smallest multiple that works. A short calculation using [6,Condition A] (done in Lemma 18.5) also shows that 2^{r−2}(t−t^{5}) ∈ R̃n^{free}(G), and it remains to consider the surgery obstruction.

For r = 3 it follows from [7,Cor.(iii)] that 2(t − t^{5}) does not give a 6–dimensional similarity. By Theorem 19.3 it follows that 2(t−t^{5}) ∉ R_{t}(G). Since the obstruction has exponent 2, we get 4(t − t^{5}) ≈_{t} 0 as claimed.

For r > 3, the assertion is that 2^{r−2}(t−t^{5}) ∈ R̃t^{free}(G) already. We prove this by induction starting with 2(t − t^{5}) in R_{n}(C(8)). Suppose that 2^{r−3}(t − t^{5}) ∈ R_{n}(H) for H = C(2^{r−1}), where r > 3. Let U_{1} = 2^{r−3}t and U_{2} = 2^{r−3}t^{5} be the free representations over H = C(2^{r−1}), and let V_{1} = 2^{r−3}t and V_{2} = 2^{r−3}t^{5} be the corresponding free representations over G = C(2^{r}). Then Ind_{H}(Δ(U_{1})/Δ(U_{2})) = 0 in H^{1}(Wh(ZG^{−})/ Wh(ZH)), since we are dividing out exactly the image of Ind_{H}. It follows from Corollary 19.4 that Ind_{H}(U_{1} − U_{2}) ∈ R_{t}(G).

However, Ind_{H}(U_{1} − U_{2}) = (V_{1} − V_{2}) + (V_{1}^{τ} − V_{2}^{τ}) where τ is the group automorphism τ(t) = t^{1+2^{r−1}}. By [7,Cor(iii)] we have 2t ≈_{t} 2t^{1+2^{r−1}} and 2t^{5} ≈_{t} 2t^{5+2^{r−1}}, whenever r ≥ 4. Therefore, 2^{r−2}t = V_{1} ⊕ V_{1} ≈ V_{1} ⊕ V_{1}^{τ} and 2^{r−2}t^{5} = V_{2} ⊕ V_{2} ≈ V_{2} ⊕ V_{2}^{τ}. □

20. The proof of Theorem D.

We first summarize our information about R_{Top}(G), obtained by putting together results from previous section.

Theorem 20.1. Let G = C(2^{r}q), with q odd, and r ≧ 2.

(i) The torsion subgroup of RTop^{free}(G) is R̃Top^{free}(G).

(ii) The rank of RTop^{free}(G) is φ(q)/2 for q > 1 (resp. rank 1 for q = 1), and the

torsion is at most 2–primary. (iii) The subgroup R̃n,Top^{free}(G) has exponent two.

Remark 20.2. In part (ii), φ(q) is the Euler function. The precise number of Z/2 summands in R̃n,Top^{free}(G) is determined by working out the conditions in Theorem C on the basis elements of R̃n^{free}(G). In cases where the conditions in Theorem C can actually be evaluated, the structure of R_{Top}(G) will thus be determined completely. □

Proof. Parts (i) and (ii) of Theorem 20.1 have already been proved in Corollary 5.2, so it remains to discuss part (iii). In fact, the assertion that R̃n,Top^{free}(G) has exponent 2 is an immediate consequence of Theorem C. To see this, suppose that (V_{1} − V_{2}) is any element in R̃n^{free}(G). The obstruction to the existence of a stable non–linear similarity V_{1} ≈_{t} V_{2} is determined by the class {Δ(V_{1})/Δ(V_{2})} in the Tate cohomology group H^{1}(Wh(ZG^{−})/ Wh(ZH)), which has exponent 2. Since the Reidemeister torsion is multiplicative, Δ(V_{1} ⊕ V_{1}) = Δ(V_{1})^{2}, and we conclude that V_{1} ⊕ V_{1} ≈_{t} V_{2} ⊕ V_{2} by Theorem C. □

More generally, it follows from our results that the structure of R̃n,Top^{free}(G) is determined by working out the criteria of Theorem C on a basis of R̃n^{free}(G). Suppose that V_{1} is stably topologically equivalent to V_{2}. Then there exists a similarity

V_{1} ⊕ W ⊕ R_{−}^{l} ⊕ R_{+}^{s} ∼_{t} V_{2} ⊕ W ⊕ R_{−}^{l} ⊕ R_{+}^{s}

where W has no R_{+} or R_{−} summands, and l, s ≧ 1. But by Corollary 11.12 we may assume that s = 1, and by Theorem 16.1 that l = 1, so we are reduced to the situation handled by Theorem C.

The proof of Theorem D. We have a basis

B = {t^{i} − t^{i+2q} | (i, 2q) = 1, i ≡ 1 (mod 4), 1 ≦ i < 4q}

for R̃^{free}(G), so it remains to work out the relations given by topological similarity. Notice that R̃h^{free}(G) = R̃n^{free}(G), and that the sum of any two elements in B lies in R̃n^{free}(G). Moreover, by Corollary 2.4 there are no 6–dimensional similarities for G.

Now suppose that V_{1} ⊕ W ⊕ R_{−} ⊕ R_{+} ∼_{t} V_{2} ⊕ W ⊕ R_{−} ⊕ R_{+}, for some complex G–representation W. Then Res_{H} V_{1} ≅ Res_{H} V_{2} since q is odd, and by Theorem C we get a similarity of the form V_{1} ⊕ R_{−}^{l} ⊕ R_{+} ∼_{t} V_{2} ⊕ R_{−}^{l} ⊕ R_{+}. But by Theorem 16.1 this implies that V_{1} ⊕ R_{−} ⊕ R_{+} ∼_{t} V_{1} ⊕ R_{−} ⊕ R_{+}. Therefore, for any element

a_{i} = (t^{i} − t^{i+2q}) ∈ B we have 2a_{i} ∉ Rt^{free}(G) but 4a_{i} ∈ Rt^{free}(G). However, in

Section 17 we determined the bounded surgery obstructions for all these elements. Since i_{∗}(u_{1,j}) = ⟨−1⟩ ∈ H^{1}(A^{×}) for all j with (j, 2q) = 1 by Lemma 17.1, there are further relations a_{1} + a_{j} ≈ a_{1} + a_{k}, or a_{1} ≈ a_{j} for all i.

It follows that a basis for R̃Top^{free}(G) is given by {a_{1} | 4a_{1} ≈ 0}. □

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Department of Mathematics & Statistics, McMaster University, Hamilton, Ont. L8S 4K1, Canada

Current address: Max Planck Institut für Mathematik, Gottfried Claren Str. 26, D-53225, Bonn, Germany

   E-mail address: ian@mpim-bonn.mpg.de

Department of Mathematical Sciences, SUNY at Binghamton, Binghamton, NY, 13901, USA

   E-mail address: erik@math.binghamton.edu