FIXED POINT THEORY AND THE K –THEORETIC TRACE

Ross Geoghegan^{∗} and Andrew Nicas^{∗∗}

August 27, 1996

Abstract. The relationship between fixed point theory and K –theory is explained, both classical Nielsen theory (versus K_{0} ) and 1–parameter fixed point theory (versus K_{1} ). In particular, various zeta functions associated with suspension flows are shown to come in a natural way as “traces” of “torsions” of Whitehead and Reidemeister type.

§1. Introduction

Consider the following facts:

1. In modern geometric topology, K –theoretic obstructions to achieving desirable geometric goals sometimes occur. Examples include the Wall finiteness obstruction [Wa], the Whitehead torsion of an h–cobordism [C], [RS] and [K], and the various Reidemeister torsions associated with a suspension flow [M], [F], [GN_{2}].

2. In recent papers [GN_{1−4}] we have explained why n–parameter Nielsen Fixed Point Theory is obstructed by an element of an n^{th} Hochschild homology group. [For n = 0, this says nothing new as the 0^{th} Hochschild homology group is naturally isomorphic to a familiar free abelian group employed in Nielsen theory.]

3. For any associative ring S with unit, there is a K –theoretic trace, Tr_{∗}: K_{∗}(S) → HH_{∗}(S),

from the K –theory of S to the Hochschild homology of S.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 55M20; Secondary 19D55, 57R80, 58F20.

∗ Partially supported by the National Science Foundation.

∗∗ Partially supported by the Natural Sciences and Engineering Research Council of Canada.

4. It sometimes happens that a geometric problem is obstructed by an element α in some K –group, while an obviously related fixed point problem is obstructed by the Hochschild homology element Tr_{∗}(α).

5. It sometimes happens that a fixed point problem is obstructed by a Hochschild homology element β which is not in the image of the trace Tr_{∗}.

6. On the other hand, it sometimes happens that β lies in the image of the trace and in this manner one is led to an interesting K –theoretic obstruction which was not previously evident. We lay out these facts in this paper in some detail. Examples of Fact 4 are given in §2

and §3, of Fact 5 in §2 and §5, and of Fact 6 in §4. While the necessary theorems have already appeared in [G_{1}] and [GN_{1−4}], they have not been previously presented from the point of view expressed here. This paper is an expanded version of the talk given by one of us (Geoghegan) at the Warsaw meeting.

§2. Classical Nielsen Theory and K_{0}

Let f: (K, v) → (K, v) be a pointed cellular map on a finite CW complex, inducing ϕ: G → G on fundamental groups. Let G_{ϕ} be the set of semiconjugacy classes with respect to ϕ, where g_{1}, g_{2} ∈ G are semiconjugate if there exists g ∈ G with g_{1} = gg_{2}ϕ(g)^{−1} . The Nielsen theory of f is encoded by the Reidemeister trace R(f, v) ∈ ℤG_{ϕ}, the free abelian group generated by G_{ϕ}. The geometric definition of R(f, v) is:

R(f, v) = ∑_{fixed point classes F} ι(f, F)Φ(F) ∈ ℤG_{ϕ}

where Φ is the well-known injection of fixed point classes to semiconjugacy classes, and ι(f, F) ∈ ℤ is the index of f at the fixed point class F (see [B] or [GN_{1}, §1(B)]). Choose an orientation for each cell of K. Let (K̃, ṽ) be the universal cover of (K, v), orient the cells of K̃ compatibly with those of K, choose one cell in each G–orbit of cells to get a free ℤG–basis for the cellular chains C_{∗}(K̃). The algebraic definition of R(f, v) is:

R(f, v) = ∑_{k≥0} (−1)^{k}q(trace(f̃_{k}: C_{k}(K̃) → C_{k}(K̃) ) ) ∈ ℤG_{ϕ}

where q: ℤG → ℤG_{ϕ} extends the function which sends each g ∈ G to its semiconjugacy class (see [GN_{1}, §1(B)]). Here, C_{∗}(K̃) is regarded as a right ℤG module, and f̃_{k} denotes the ℤG–matrix whose (i, j)^{th} entry is the coefficient of the i^{th} k –cell in the image of the j^{th} k –cell (considered as basis elements); see [GN_{1}, §1(B)].

The assertion that these two definitions are equivalent is essentially due to Reidemeister [R] and Wecken [W]; see, for example, [G_{2}] for a proof.

In this section we will assume that ϕ is the identity endomorphism 1: G → G, and G_{1} will denote the set of conjugacy classes of G. So R(f, v) ∈ ℤG_{1} .

We recall the definition of K_{0}(ℤG) and K̃_{0}(ℤG). Denote the isomorphism class of a finitely generated projective ℤG–module P by [P]. The abelian group K_{0}(ℤG) is defined to be the free abelian group generated by all such [P] modulo the relations [P] + [Q] = [P ⊕ Q]. Let {P } be the corresponding element of K_{0}(ℤG). There is a natural homomorphism ℤ → K_{0}(ℤG) taking 1 to {ℤG}. The cokernel, K̃_{0}(ℤG), is the reduced projective class group of the ring ℤG. We write π: K_{0}(ℤG) → K̃_{0}(ℤG) for the quotient homomorphism. For us, the only relevant property of K̃_{0}(ℤG) is distinguishing zero from non-zero elements: π({P }) = 0 if and only if P is stably free, i.e. the direct sum of P with some finitely generated free module is a free module. See, for example, [Br, p.201].

If A is an idempotent m × m ℤG–matrix, the image of A: ℤG^{m} → ℤG^{m} is a projective ℤG–module; conversely if P is a finitely generated projective ℤG–module it is a direct summand of a free module ℤG^{m}, and the projection of ℤG^{m} onto the submodule P of ℤG^{m} is represented by an idempotent matrix. So elements of K_{0}(ℤG) can be regarded as equivalence classes {A} of idempotent ℤG–matrices A. The homomorphism A ↦ trace(A) ↦ q(trace(A)) well-defines the K –theoretic (or Hattori-Stallings) trace Tr_{0} : K_{0}(ℤG) → ℤG_{1} . For a proof that this is well-defined see [St, §1].

When is R(f, v) in the image of Tr_{0} ? A partial answer is: When f is a homotopy idempotent, i.e. f is homotopic to f^{2} , as we now explain.

Let f be a homotopy idempotent. We say that f splits if there exist a (not necessarily finite) CW complex X and maps d: K → X and u: X → K such that d ∘ u ≃ id_{X} and u ∘ d ≃ f. In that case X is a finitely dominated complex (the existence of K, u and d with K finite and d ∘ u ≃ id_{X} being the definition). We may assume (giving X a basepoint) that u and d are basepoint preserving maps. It is a straightforward theorem that if f ≃ f^{2} rel v then f splits. A much harder theorem of Hastings and Heller asserts that the “rel v ” is unnecessary:

Theorem [HH]. If f is a homotopy idempotent on a finite dimensional CW complex K then f splits.

For another proof see [BG].

So a homotopy idempotent f: (K, v) → (K, v) always gives rise in this way to a finitely dominated complex X; indeed X is unique up to homotopy equivalence. Conversely, given X, a homotopy idempotent f = u ∘ d: (K, v) → (K, v) can always be chosen which splits through X and satisfies f_{#} = 1: G → G. Finally, if f′: (K′, v′) → (K′, v′) is another homotopy idempotent splitting through X then R(f, v) and R(f′, v′) encode the same information (same Nielsen numbers, same indices, canonically corresponding fixed point classes). For more on all this, see [G_{1}].

In view of what has been said, we will assume in what follows that u and d induce isomorphisms on fundamental groups and we will identify π_{1}(K, v) with π_{1}(X, x) via d_{#} , calling both groups G.

Next, we discuss the Wall obstruction σ(X) ∈ K̃_{0}(ℤG) of a finitely dominated space X whose fundamental group is G. This is defined in [Wa]; for a helpful relevant discussion, see [Br, ch. 8], especially page 201.

Theorem [Wa]. σ(X) = 0 if and only if X has the homotopy type of a finite complex.

Choose A so that π({A}) = σ(X), and consider Tr_{0}({A}) ∈ ℤG_{1} . If I is a k × k identity matrix then π({A}) = π({A ⊕ I}) and Tr_{0}({A ⊕ I}) = Tr_{0}({A}) + k[1], so the sum of the coefficients in Tr_{0}({A}) is not determined by σ(X), but can be any integer. The following theorem is essentially proved in [G_{1}]:

Theorem 2.1. If π({A}) = σ(X) and A is chosen so that the sum of the coefficients in Tr_{0}({A}) is the Euler characteristic χ(X) then R(f, v) = Tr_{0}({A}) ≡ q(trace(A)). □

In summary, the Reidemeister trace of a homotopy idempotent on a finite complex always lies in the image of the K –theoretic trace Tr_{0} : K_{0}(ℤG) → ℤG_{1} .

The Strong Bass Conjecture [Ba] asserts that the image of this trace is always a multiple of the trivial conjugacy class [1] ∈ G_{1} ⊂ ℤG_{1} . This conjecture is known to be valid for some large classes of groups (see [G_{2}] for references). When this holds for G, it follows that for any map f: (K, v) → (K, v) (not necessarily a homotopy idempotent) inducing the identity on the fundamental group G, R(f, v) is in the image of the trace if and only if the Nielsen number N(f) = 0 or 1 and when N(f) = 1 the only essential fixed point class is that of the basepoint.

When f does not induce the identity on the fundamental group (and is not “equivalent” to a map g which does, i.e. f = r ∘ s and g = s ∘ r) it is not clear to us if K_{0} of some other ring might replace K_{0}(ℤG) giving R(f, v) in the image of some sort of K –theoretic trace.

Finally, we remark that from an algebraic point of view, it is useful to view ℤG_{1} as the 0^{th} Hochschild homology group HH_{0}(ℤG) of the ring ℤG. This could be avoided up to now, but must be confronted in the next section.

§3. The s–Cobordism Theorem and a related one-parameter fixed point problem

In this section we recall 1–parameter Nielsen theory and the s–Cobordism Theorem, and we relate the two via the K –theoretic trace.

(A) One-parameter Nielsen fixed point theory.

Let F: M × I → M be a homotopy where M is a compact oriented manifold. We consistently work in the smooth or PL category. Pick a basepoint v ∈ M. Identify π_{1}(M × I, (v, 0)) with G ≡ π_{1}(M, v) using the isomorphism induced by the projection map p: M × I → M. Define Fix(F) = {(x, t) ∈ M × I | F (x, t) = x}; its elements are the fixed points of F.

Assume F (v, 0) = v and that F induces the identity 1: G → G. Two fixed points (x, t) and (y, t′) of F are in the same fixed point class if for some path ν from (x, t) to (y, t′), the loop (p ∘ ν)(F ∘ ν)^{−1} is homotopically trivial. This defines an equivalence relation on Fix(F). Just as in the classical case, there is an injective function Ψ from the set of fixed point classes of F to the set, G_{1} , of conjugacy classes: the class containing (x, t) is mapped to the conjugacy class C containing g_{C} ≡ [(p ∘ μ)(F ∘ μ)^{−1}], where μ is any path from the basepoint (v, 0) to (x, t). It is easy to check that Ψ is well–defined and injective, that F has only finitely many fixed point classes, and that fixed points in the same path component of Fix(F) are in the same fixed point class. Moreover, with the same notation, let ω be a loop in Fix(F) ⊂ M × I based at (x, t), and let h = [μωμ^{−1}] ∈ G; then h lies in the centralizer, Z(g_{C}), of g_{C}.

By transversality, we can perturb F rel M ×{0, 1} so that Fix(F) consists of transverse circles in the interior of M × I which, for all but finitely many values of t, cross M × {t} transversely, and other path components of fixed points each of which meets M × {0, 1}. If (x, t) lies in such a circle crossing M × {t} transversely, orient the circle in the direction of positive time if ι(F_{t}, x) is positive, and in the other direction if it is negative; see [DG,§8 and §11] for details.

We are only interested in the circles, indeed in circles not in the same fixed point class as any fixed points in M × {0, 1}.^{1} Let V be such a circle, let (x, t) ∈ V , let μ be a path in M × I from (v, 0) to (x, t), and let ω be the loop based at (x, t) obtained by traversing V once in the direction of its orientation. With notation as above, we have, as previously indicated, that the corresponding element, h, of G lies in Z(g_{C}). In this way, we associate with V an element of H_{1}(Z(g_{C})) ≅ (Z(g_{C}))_{ab} . If there are two circles V_{1} and V_{2} in the same fixed point class, we reach the same centralizer Z(g_{C}) from both circles provided the path used for (x_{1}, t_{1}) ∈ V_{1} is μ, and the path used for (x_{2}, t_{2}) ∈ V_{2} is μν, where p(ν)F (ν)^{−1} is homotopically trivial; we treat any (finite) number of circles similarly. Thus, for each fixed point class, F, of F which does not meet M × {0, 1}, we have defined an element ι(F, F) ∈ H_{1}(Z(g_{C})), where g_{C} ≡ [p(μ)F (μ)^{−1}] represents the conjugacy class C, and Ψ(F) = C.

1The reason for this is that in the remaining fixed point classes the distinction between circles and other fixed points is not invariant under homotopy rel M × {0, 1}. For example, a transverse arc of fixed points beginning and ending in M × {0, 1} might merge with a transverse circle in the same fixed point class after a homotopy rel M × {0, 1}.

The transverse intersection invariant of F is:

R(F, v) = ∑_{j=1}^{k} ι(F, F_{j}) ∈ ⨁_{C∈G_{1}} H_{1}(Z(g_{C}))

where F_{1}, …, F_{k} are the fixed point classes of F which do not meet M × {0, 1}. Note that since F_{0}(v) = v, the fixed point class of v is not among the F_{j} ’s.

Now, ⨁_{C∈G_{1}} H_{1}(Z(g_{C})) should remind the reader of

⨁_{C∈G_{1}} H_{0}(Z(g_{C})) ≅ ⨁_{C∈G_{1}} ℤ ≅ ℤG_{1}

in §2; these direct sums have the following algebraic reformulation.

The Hochschild homology groups of an associative ring with unit, S, are the homology groups of a certain chain complex which in degrees 2, 1, 0 is given by:

⋯ ⟶ S ⊗ S ⊗ S ⟶^{d_{2}}  S ⊗ S ⟶^{d_{1}}  S
s_{1} ⊗ s_{2} ⊗ s_{3} ↦  s_{2} ⊗ s_{3}s_{1} − s_{1}s_{2} ⊗ s_{3} + s_{1} ⊗ s_{2}s_{3}
s_{1} ⊗ s_{2} ↦ s_{2}s_{1} − s_{1}s_{2}.

In particular, HH_{0}(S) = coker(d_{1}) and HH_{1}(S) = ker(d_{1})/ image(d_{2}). When S = ℤG there are isomorphisms HH_{i}(ℤG) ≅ ⨁_{C∈G_{1}} H_{i}(Z(g_{C})) for all i ≥ 0 (see [GN_{1−4}] for a detailed description). These isomorphisms are canonical once the representatives g_{C} of the conjugacy classes C have been chosen. This suggests that it may be useful to regard R(F, v) as an element of HH_{1}(ℤG).

(B) K_{1} , the Whitehead group, and Whitehead torsion.

Let S be a ring with unit, let GL(n, S) denote the general linear group consisting of all n × n invertible matrices over S, and let GL(S) be the direct limit of the sequence GL(1, S) ⊂ GL(2, S) ⊂ · · · . A matrix in GL(S) is called elementary if it coincides with the identity except for a single off-diagonal entry. The subgroup E(S) ⊂ GL(S) generated by the elementary matrices is precisely the commutator subgroup of GL(S) (see [C]) and the abelian quotient group GL(S)/E(S) is, by definition, K_{1}(S). If A ∈ GL_{n}(S) the image of A in K_{1}(S) is called the torsion of A.

Now let U ⊂ S^{∗} be a subgroup of the group of units of S. Assume S has the property that any two bases of a finitely generated free module have the same cardinality. Declare two bases of a finitely generated free module to be equivalent if the change of basis matrix represents an element of K_{1}(S) which lies in the image of the natural map U → K_{1}(S). Define K_{1}^{U}(S) ≡ coker(U → K_{1}(S)). Let (C, ∂) be a finitely generated chain complex of right modules over S such that each C_{i} is free with a given equivalence class of bases. Suppose that C is acyclic. Let δ: C → C be a chain contraction. Define C_{odd} ≡ ⨁_{i odd} C_{i} and C_{even} ≡ ⨁_{i even} C_{i}. The restriction of ∂ +δ to C_{odd} is an isomorphism C_{odd} → C_{even} and so its matrix with respect to bases chosen from each of the given equivalence classes defines an element of K_{1}(S). The image of this element in K_{1}^{U}(S) is independent of the choice of representatives of the equivalence classes of bases (see [C]); it is called the torsion of (C, ∂) with respect to U. The important case for the purpose of this section is S = ℤG and U = ±G. In this case Wh(G) ≡ K_{1}^{±G}(ℤG) is called the Whitehead group of G and the torsion of an acyclic complex (C, ∂) as above is called its Whitehead torsion.

An h–cobordism (M; M_{0}, M_{1}) is a manifold M with two boundary components M_{0} and M_{1} both of which are strong deformation retracts of M. The torsion of (M; M_{0}, M_{1}) is the Whitehead torsion, τ(M), of the relative cellular chain complex C_{∗}(M̃, M̃_{0}) with respect to a CW structure on the triad (M; M_{0}, M_{1}). In [RS] one finds a proof of:

Theorem (s-Cobordism Theorem). If dim M ≥ 6 and M is compact, (M; M_{0}, M_{1}) is PL homeomorphic to (M_{0} × I, M_{0} × {0}, M_{0} × {1}) if and only if τ(M) = 0.

The same theorem is true in the smooth category, see [K].

(C) The K –theoretic trace.

The K –theoretic trace (or Dennis trace) Tr_{1} : K_{1}(S) → HH_{1}(S) is defined as follows. If α ∈ K_{1}(S) is represented by an invertible n × n matrix A, then Tr_{1}(α) is represented by the Hochschild 1–cycle ∑_{i,j} A_{ij} ⊗ (A^{−1})_{ji}.

Consider the diagram

K_{1}(ℤG) ⟶^{Tr_{1}} HH_{1}(ℤG) ≅ ⨁_{C∈G_{1}} H_{1}(Z(g_{C}))
π↓
Wh(G)

where π is the obvious quotient. The feature to note is:

Proposition 3.1. If α, β ∈ K_{1}(ℤG) are such that π(α) = π(β) then Tr_{1}(α) − Tr_{1}(β) can only be non-zero in the component of the trivial conjugacy class [1] ∈ G_{1} . □

(D) The K –theoretic trace and Nielsen fixed point theory.

Let (M; M_{0}, M_{1}) be an h–cobordism with M compact. Let F: M × I → M be a strong deformation retract of M onto M_{0} . Then we can perturb F rel M × {0, 1} (using relative transversality and ad hoc methods) so that Fix(F) = (M × {0}) ∪ (M_{0} × I) ∪ transverse circles as described in (B). Now, (M × {0}) ∪ (M_{0} × I) as well as, perhaps, some circles are in the fixed point class of the basepoint v. We are only interested in the other circles, namely the ones that make up the components of R(F, v). Those are the ones that cannot disappear under perturbation of F rel M × {0, 1}.

Here is the main theorem of this section.

Theorem 3.2. If π(α) = τ(M) then the projection of Tr_{1}(α) away from the component of the trivial conjugacy class is equal to R(F, v). □

In other words, apart from an ambiguity over the trivial conjugacy class (analogous to the ambiguity in getting {A} from σ(X) in Theorem 2.1), the one-parameter fixed point problem contained in our setup is the “trace problem” associated with the h–cobordism problem. Note that if M is homeomorphic to M_{0} × I and if F is the “projection” (i.e. F ((y, u), t) = (y, ut) where (y, u) ∈ M_{0} × I) then Fix(F) = M × {0} ∪ M_{0} × I, so R(F, v) = 0.

Proposition 3.1 and Theorem 3.2 are proved in [GN_{1}, §7].

§4. Torsion and zeta functions

Let f: (K, v) → (K, v) be a pointed cellular map on a finite CW complex, inducing ϕ: G → G on fundamental groups. In this section we do not assume that ϕ is the identity. A periodic point of period m is a fixed point of the m–fold iterate f^{m}. If F is a fixed point class of the map f^{m}, so are f(F), …, f^{m−1}(F), and the corresponding m–periodic orbit class of f is ⋃_{j=0}^{m−1} f^{j}(F), where f^{0}(F) ≡ F. Note that the fixed point classes f^{i}(F) and f^{j}(F) are either identical or disjoint.

One classifies m–periodic orbit classes algebraically thus: the relation g ∼ ϕ(g) welldefines an equivalence relation on the set G_{ϕ^{m}}; write G_{ϕ^{m}}/⟨ϕ⟩ for the set of equivalence classes. By analogy with Φ in §2, there is an injective function Φ′ from the set of m–periodic orbit classes of f to the set G_{ϕ^{m}}/⟨ϕ⟩. Define

R̄(f^{m}, v) = ∑_{m–periodic orbit classes O} ι(f^{m}, O)Φ′(O) ∈ ℤ[G_{ϕ^{m}}/⟨ϕ⟩].

The obvious quotient homomorphism takes R(f^{m}, v) to R̄(f^{m}, v), the effect being to coalesce the distinct fixed point classes of f^{m} which are in the same m–periodic orbit class of f, adding up the fixed point indices of the distinct pieces. The Lefschetz-Nielsen series of f is the sequence (R̄(f^{m}, v)) ∈ ∏_{m≥1} ℤ[G_{ϕ^{m}}/⟨ϕ⟩]; we will see in Theorem 4.1 that the Lefschetz-Nielsen series is the “trace” of an interesting element of K –theory and that this fact implies the rationality of various zeta functions associated with the map f.

Let ℤG^{ϕ}[[t]] denote the ϕ–twisted power series ring over ℤG, consisting of formal power series ∑_{m≥0} u_{m}t^{m} in the indeterminate t, with u_{m} ∈ ℤG, subject to the multiplication rule tg = ϕ(g)t where g ∈ G. If A is a square ℤG–matrix and I is the identity matrix of the same size then the matrix I − At is invertible in the ring of ℤG^{ϕ}[[t]]– matrices; its inverse is ∑_{m≥0}(At)^{m}. This simple observation turns out to be important. Define Δ(f, v) ∈ K_{1}(ℤG^{ϕ}[[t]]) to be the element represented by the invertible matrix c(f) ≡ ∏_{k≥0}(I − f̃_{k}t)^{(−1)^{k+1}} . The matrix c(f) depends on choices (of lifts, orientations etc., see §2) but its torsion, Δ(f, v), does not [GN_{2}, Proposition 5.2].

We have a trace Tr_{1}(Δ(f, v)) ∈ HH_{1}(ℤG^{ϕ}[[t]]), but to obtain the the Lefschetz-Nielsen series, we must compose this with a certain canonical homomorphism

P_{+} : HH_{1}(ℤG^{ϕ}[[t]])) → ∏_{m≥1} ℤ[G_{ϕ^{m}}/⟨ϕ⟩]

which we now define.

For n ≥ 0, let C_{1,n} be the abelian subgroup of ℤG^{ϕ}[[t]] ⊗ ℤG^{ϕ}[[t]] (the group of Hochschild 1–chains on ℤG^{ϕ}[[t]]) generated by tensors of the form yt^{a} ⊗ zt^{b} where y, z ∈ ℤG and a+b = n, and let C_{2,n} be the abelian subgroup of ℤG^{ϕ}[[t]]⊗ℤG^{ϕ}[[t]]⊗ℤG^{ϕ}[[t]] (the group of Hochschild 2–chains on ℤG^{ϕ}[[t]]) generated by tensors of the form ut^{a} ⊗vt^{b} ⊗wt^{c} where u, v, w ∈ ℤG and a + b + c = n. Note that the Hochschild boundary operator d_{2} (see §3(A)) maps C_{2,n} into C_{1,n}. Define

HH_{1,n}(ℤG^{ϕ}[[t]]) ≡ ker(d_{1} : C_{1,n} → ℤG^{ϕ}[[t]])/ image(d_{2} : C_{2,n} → C_{1,n}).

The homomorphism ℤG^{ϕ}[[t]] ⊗ ℤG^{ϕ}[[t]] → ∏_{n=0}^{∞} C_{1,n} given by

(∑_{i=0}^{∞} y_{i} t^{i}) ⊗ (∑_{j=0}^{∞} z_{j} t^{j}) ↦ ( ∑_{i+j=n} y_{i} t^{i} ⊗ z_{j} t^{j})_{n=0}^{∞},

where y_{i}, z_{j} ∈ ℤG induces a homomorphism

HH_{1}(ℤG^{ϕ}[[t]]) ⟶ ∏_{n≥0} HH_{1,n}(ℤG^{ϕ}[[t]]).

Define a homomorphism J_{n}: C_{1,n} → ℤG as follows. Given a generating 1–chain yt^{a} ⊗ zt^{b} ∈ C_{1,n}, let J_{n}(yt^{a} ⊗ zt^{b}) = a yϕ^{a}(z). Suppose β = ut^{a} ⊗ vt^{b} ⊗ wt^{c} ∈ C_{2,n}. Then J_{n}(d_{2}β) = b( vϕ^{b}(w)ϕ^{b+c}(u) − uϕ^{a}(v)ϕ^{a+b}(w) ). Since vϕ^{b}(w)ϕ^{b+c}(u) − uϕ^{a}(v)ϕ^{a+b}(w) maps to zero in ℤ[G_{ϕ^{n}}/⟨ϕ⟩], it follows that J_{n} induces a homomorphism:

j_{n}: HH_{1,n}(ℤG^{ϕ}[[t]]) → ℤ[G_{ϕ^{n}}/⟨ϕ⟩].

Define P_{+} : HH_{1}(ℤG^{ϕ}[[t]]) ⟶ ∏_{n=1}^{∞} ℤ[G_{ϕ^{n}}/⟨ϕ⟩] to be the composite:

HH_{1}(ℤG^{ϕ}[[t]]) ⟶ ∏_{n≥0} HH_{1,n}(ℤG^{ϕ}[[t]]) ⟶^{proj} ∏_{n≥1} HH_{1,n}(ℤG^{ϕ}[[t]]) \underrightarrow{∏_{n} j_{n}} ∏_{n=1}^{∞} ℤ[G_{ϕ^{n}}/⟨ϕ⟩].

The image of Δ(f, v) under Tr_{1} is the element of HH_{1}(ℤG^{ϕ}[[t]]) represented by the Hochschild 1–cycle:

∑_{k≥0}(−1)^{k+1} ∑_{i,j} (I − f̃_{k}t)_{ij} ⊗ ( ∑_{m≥0}(f̃_{k}t)^{m})_{ji}.

More importantly,

Theorem 4.1. The image of Δ(f, v) under P_{+} ∘ Tr_{1} : K_{1}(ℤG^{ϕ}[[t]]) → ∏_{m≥1} ℤ[G_{ϕ^{m}}/⟨ϕ⟩] is the Lefschetz-Nielsen series of f, (R̄(f^{m}, v)). □

This is Theorem 5.6 of [GN_{2}].

To relate this to zeta functions, consider a linear representation ρ: G → GL_{r}(S) where S is a commutative ring with unit and ρ ∘ ϕ = ρ. The ρ–twisted Lefschetz number of f is defined to be the image, L(f, ρ), of R(f, v) under the homomorphism induced by ρ. In detail, ρ induces ρ̄: ℤG → M_{r}(S), the ring of r × r matrices over S, and hence ρ_{∗} ≡ trace ∘ρ̄: ℤG → S. Define L(f, ρ) = ∑_{k≥0}(−1)^{k} trace(ρ_{∗}(f̃_{k})) ∈ S, where ρ_{∗} maps ℤG–matrices to S –matrices entry by entry. The homomorphism ρ induces a function G_{ϕ} → GL_{r}(S) because ρ ∘ ϕ = ρ, so L(f, ρ) coalesces the components of

R(f, v) = ∑_{fixed point classes F} ι(f, F)Φ(F)

to give first an S –matrix, and then an element of S, namely the trace of that S –matrix.

The representation ρ induces a homomorphism K_{1}(ℤG^{ϕ}[[t]]) → K_{1}(S[[t]]). The image of Δ(f, v) is Δ(f, ρ) represented by c(f, ρ) ≡ ∏_{k≥0}(I − ρ_{∗}(f̃_{k})t)^{(−1)^{k+1}} . Applying the determinant homomorphism det: K_{1}(S[[t]]) → S[[t]] we obtain

det Δ(f, ρ) = ∏_{k≥0} det(I − ρ_{∗}(f̃_{k})t)^{(−1)^{k+1}} ∈ S[[t]].

Define the ρ–twisted zeta function, ζ(f, ρ) ∈ S[[t]], to be exp ( ∑_{m=1}^{∞} \frac{1}{m}L(f^{m}, ρ)t^{m}). Then “rationality of the zeta function” is expressed by:

Theorem 4.2. det Δ(f, ρ) = ζ(f, ρ).

Theorem 4.2 is a direct consequence of Theorem 4.1. The homomorphism ρ̄ induces

ℤ[G_{ϕ^{m}}/⟨ϕ⟩] ⟶ M_{r}(S) ⟶^{trace} S

which maps R̄(f^{m}, v) to L(f^{m}, ρ); in fact, it induces a homomorphism

K_{1}(ℤG^{ϕ}[[t]]) ⟶ ∏_{m≥1} ℤ[G_{ϕ^{m}}/⟨ϕ⟩] ⟶ S[[t]]

which maps Δ(f, v) ↦ (R̄(f^{m}, v)) ↦ ζ(f, ρ). Theorem 4.2 follows from the fact that this homomorphism equals Δ(f, v) ↦ Δ(f, ρ) ↦ det Δ(f, ρ). The details can be found in [GN_{2}, §5(B)].

Note that when S = ℤ, r = 1 and ρ(G) = {1} ∈ ℤ then L(f, ρ) = L(f), the Lefschetz number of f, and ζ(f, ρ) is the familiar Lefschetz zeta function exp ( ∑_{m=1}^{∞} \frac{1}{m}L(f^{m})t^{m}) of Milnor [M]. Other choices give the zeta functions appearing in [GN_{2}], [J], and [F].

In view of this, we regard the Lefschetz-Nielsen series as a “non-abelian” analogue of the zeta function. The invariant Δ(f, v) was introduced by us in [GN_{2}] and is an example of Fact (6) in the Introduction. We discovered it through our study of Tr_{1}(Δ(f, v)). This is explained in [GN_{2}] where we also show that Δ(f, v) depends only on the homotopy class of f and on the simple homotopy type of K.

§5. First-order Euler characteristics

(A) More on one-parameter Nielsen fixed point theory.

Let F: K × I → K be a cellular homotopy where K is a finite CW complex and F_{0} = F_{1} = id_{K}. Just as in §3(A), we set G ≡ π_{1}(K, v) and identify G with π_{1}(K × I, (v, 0)); here, v is a vertex of K. And just as in §2, we consider the cellular chains C_{∗}(K̃) and C(K): F induces a chain homotopy {D_{k}: C_{k}(K) → C_{k+1}(K)}, and F̃ induces a chain homotopy {D̃_{k}: C_{k}(K̃) → C_{k+1}(K̃)}. Precisely: D̃_{k}(ẽ) = (−1)^{k+1}F̃_{k}(ẽ × I) and D_{k}(e) = (−1)^{k+1}F_{k}(e × I) where e is any k –cell of K (with a chosen orientation as in §2).^{2}

We also have the boundary maps {∂_{k}: C_{k}(K) → C_{k−1}(K)} and {˜∂_{k}: C_{k}(K̃) → C_{k−1}(K̃)}. If g_{0} ∈ G is represented by the loop F ({v} × I) then F̃_{1}(ẽ) = ẽg_{0}^{−1} ; here g_{0}^{−1} appears rather than g_{0} because the natural left action of G has been turned into a right action.

The chain homotopy relations give matrix equations:

D̃_{k−1} ˜∂_{k} + ˜∂_{k+1}D̃_{k} = I(1 − g_{0}^{−1}) and D_{k−1}∂_{k} + ∂_{k+1}D_{k} = 0

where I is a suitable identity matrix.

It is convenient to write D̃ ≡ ⨁_{k}(−1)^{k+1}D̃_{k} and ˜∂ ≡ ⨁_{k} ˜∂_{k}, with similar definitions for D and ∂. Then

trace(D̃ ˜∂ − ˜∂D̃) = (1 − g_{0}^{−1})χ(K) and D∂ − ∂D = 0 where χ(K) is the Euler characteristic of K.

2We use the following sign convention. If ẽ is an oriented k –cell of K̃ then D̃_{k}(ẽ) is the (k + 1)–chain (−1)^{k+1}F̃_{∗}(ẽ × I) ∈ C_{k+1}(K̃), where ẽ × I is given the product orientation. This is consistent with the convention that if E_{i,ϵ} is the face of the cube I^{n} = [0, 1]^{n} obtained by holding the i^{th} coordinate fixed at ϵ = 0 or 1, then the incidence number [I^{n} : E_{i,ϵ}] is (−1)^{i+ϵ}. At the level of cellular n–chains, we have ∂_{n}I^{n} = ∑_{i,ϵ}[I^{n} : E_{i,ϵ}]E_{i,ϵ}.

Proposition 5.1. The Hochschild 1–chain ∑_{i,j} ˜∂_{ij} ⊗ D̃_{ji} is a cycle.

Proof. (Compare [GN_{3}, Proposition 2.4].) The image of ∑_{i,j} ˜∂_{ij}⊗D̃_{ji} under the Hochschild boundary (d_{1} of §3(A)) is (1−g_{0}^{−1})χ(K). If χ(K) = 0 the conclusion follows. At any rate (1 − g_{0}^{−1})χ(K) is a boundary, so it represents zero in HH_{0}(ℤG), and hence if χ(K) ≠ 0 we must have g_{0}^{−1} = 1. □

(B) First-order Euler characteristics.

Let Γ_{K} ≡ π_{1}(K^{K}, id). Thus γ ∈ Γ_{K} is represented by F^{γ}: K × I → K as in (A). We write D̃^{γ} and D^{γ} for the corresponding chain homotopies. By Proposition 5.1 we may define a function X̃_{1}(K): Γ_{K} ⟶ HH_{1}(ℤG) by

X̃_{1}(K)(γ) = homology class of ∑_{i,j} ˜∂_{ij} ⊗ D̃_{ji}^{γ}.

There is a left action of Γ_{K} on HH_{1}(ℤG) given at the level of chains by γ(g_{1} ⊗ g_{2}) = g_{1} ⊗ g_{2}η(γ)^{−1} ; here, g_{1}, g_{2} ∈ G and η: Γ_{K} → G takes γ to the element of G represented by the loop F^{γ}(v, · ). The image of η is called the Gottlieb subgroup of π_{1}(K, v) and lies in the center, Z(G), of G.

Thus we may regard HH_{1}(ℤG) as a left Γ_{K} –module, and it is easy to see that:

Proposition 5.2. The function X̃_{1}(K) is a derivation i.e., X̃_{1}(K)(γ_{1}γ_{2}) = X̃_{1}(K)(γ_{1}) + γ_{1}X̃_{1}(K)(γ_{2}). □

If M is a left Γ_{K} –module then the cohomology group H^{1}(Γ_{K}, M) is naturally identified with the quotient, Der(Γ_{K}, M)/ Inn(Γ_{K}, M), of derivations modulo inner derivations (recall that a derivation Γ_{K} → M is inner if it is of the form γ ↦ (1 − γ)m for some m ∈ M). We denote the image of a derivation Θ: Γ_{K} → M in H^{1}(Γ_{K}, M) by [Θ].

The enhanced first-order Euler characteristic of K is the cohomology class:

χ̃_{1}(K) ≡ [X̃_{1}(K)] ∈ H^{1}(Γ_{K}, HH_{1}(ℤG)).

To explain why we call this invariant “enhanced” we digress to define a simpler invariant with which χ̃_{1}(K) should be compared. The homology of G with integer coefficients can be computed from a chain complex which in degrees 2, 1, 0 is:

⋯ ⟶ ℤG ⊗ ℤG ⟶^{d′_{2}} ℤG ⟶^{d′_{1}} ℤ

where d′_{2}(g_{1}⊗g_{2}) = g_{2}−g_{1}g_{2}+g_{1} for g_{1}, g_{2} ∈ G and d′_{1}(g) = 1 for g ∈ G. [This is obtained from the Hochschild complex of §3(A) by replacing the rightmost tensor factor of ℤG by ℤ, i.e. H_{∗}(G) ≅ HH_{∗}(ℤG, ℤ) where HH_{∗}(ℤG, ℤ) is Hochschild homology with coefficients in the trivial bimodule ℤ.] Let A: ℤG → G_{ab} extend the “abelianization” homomorphism G → G_{ab} ≡ H_{1}(G). There is a natural homomorphism ε: HH_{1}(ℤG) → H_{1}(G) defined by ε({∑_{i} n_{i}g_{1,i} ⊗ g_{2,i}}) = A(∑_{i} n_{i}g_{1,i}) where ∑_{i} n_{i}g_{1,i} ⊗ g_{2,i} is a Hochschild cycle and {·} denotes Hochschild homology class. The homomorphism ε induces:

ε_{∗}: H^{1}(Γ_{K}, HH_{1}(ℤG)) ⟶ H^{1}(Γ_{K}, H_{1}(G)) ≅ Hom(Γ_{K}, H_{1}(G)).

In [GN_{3}] we defined the first-order Euler characteristic of K to be the homomorphism χ_{1}(K): Γ_{K} → H_{1}(G) given by χ_{1}(K) = A(∑_{i,j} ˜∂_{ij}D_{ji}). The comparison with χ̃_{1}(K) is: Proposition 5.3. ε_{∗}(χ̃_{1}(K)) = χ_{1}(K). □

These first order Euler characterisitics appear to be useful new invariants. Applications of χ_{1}(K) to group theory are given in [GN_{3}] and to geometry in [GNO], while applications of χ̃_{1}(K) to differential topology and to 3–dimensional Seifert fibered spaces are given in [GN_{4}]. Remark. The invariant χ_{1}(K) is a natural object from the point of view of algebraic topology; over a field of coefficients 𝔽, χ_{1}(K; 𝔽) can be expressed homologically in terms of cap (or cup) products (see [GN_{3}]) and higher analogs χ_{n}(K; 𝔽), n > 1 can be defined and calculated [GNO]. (C) The trace.

Here is a simple observation (see [GN_{4}, Proposition 1.4] ):

Proposition 5.4. The image of the composite homomorphism:

±G ⟶ K_{1}(ℤG) ⟶^{Tr_{1}} HH_{1}(ℤG)

lies in the component of HH_{1}(ℤG) ≅ ⨁_{C∈G_{1}} H_{1}(Z(g_{C})) corresponding to the trivial conjugacy class [1]. □

Proposition 5.4 implies that when Wh(G) = 0 the image of Tr_{1} : K_{1}(ℤG) → HH_{1}(ℤG) is concentrated in the component corresponding to the trivial conjugacy class. On the other hand, in [GN_{4}, §6] we give examples of finite complexes K which have the following properties:

(1) K triangulates a closed aspherical 3–manifold.

(2) Wh(G) = 0.

(3) K admits the structure of an oriented Seifert fibered space and the element γ ∈ Γ_{K} given by the S^{1} –action on K associated to this Seifert fibered space structure is such that X̃_{1}(K)(γ) is not concentrated in the component of the trivial conjugacy class.

Moreover, in these examples the third property persists after passage to the cohomology class χ̃_{1}(K). More precisely, Tr_{1} : K_{1}(ℤG) → HH_{1}(ℤG) extends to a Γ_{K} –module homomorphism Tr′_{1} : ℤΓ_{K} ⊗ K_{1}(ℤG) → HH_{1}(ℤG) and χ̃_{1}(K) is not in the image of the induced homomorphism (Tr′_{1})_{∗}: H^{1}(Γ_{K}, ℤΓ_{K} ⊗ K_{1}(ℤG)) → H^{1}(Γ_{K}, HH_{1}(ℤG)).

References

[Ba] H. Bass, Euler characteristics and characters of discrete groups, Invent. Math. 35 (1976), 155– 196.

[B] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott Foresman, Chicago, 1971.

[Br] K. S. Brown, Cohomology of Groups, Springer–Verlag, New York, 1982.

[BG] K. S. Brown and R. Geoghegan, An infinite dimensional torsion-free F P_{∞} group, Invent. Math. 77 (1984), 367–381.

[C] M. M. Cohen, A Course in Simple–Homotopy Theory, Springer–Verlag, New York, 1973.

[DG] D. Dimovski and R. Geoghegan, One-parameter fixed point theory, Forum Math. 2 (1990), 125– 154.

[F] D. Fried, Homological identities for closed orbits, Invent. Math. 71 (1983), 419–442.

[G_{1} ] R. Geoghegan, Fixed points in finitely dominated compacta: the geometric meaning of a conjecture of H. Bass, Shape Theory and Geometric Topology, Lecture Notes in Math. Vol. 870, Springer-Verlag, New York, 1981, pp. 6–22.

[G_{2} ] R. Geoghegan, Nielsen fixed point theory, Handbook of Geometric Topology, (to be published by

Elsevier).

[GN_{1} ] R. Geoghegan and A. Nicas, Parametrized Lefschetz–Nielsen fixed point theory and Hochschild

homology traces, Amer. J. Math. 116 (1994), 397–446.

[GN_{2} ] R. Geoghegan and A. Nicas, Trace and torsion in the theory of flows, Topology 33 (1994),

683–719.

[GN_{3} ] R. Geoghegan and A. Nicas, Higher Euler characteristics (I), L’Enseign. Math. 41 (1995), 3–62. [GN_{4} ] R. Geoghegan and A. Nicas, A Hochschild homology Euler characteristic for circle actions,

preprint.

[GNO] R. Geoghegan, A. Nicas and J. Oprea, Higher Lefschetz Traces and spherical Euler characteristics, Trans. Amer. Math. Soc. 348 (1996), 2039–2062.

[HH] H. M. Hastings and A. Heller, Homotopy idempotents on finite dimensional complexes split, Proc. Amer. Math. Soc. 85 (1982), 619–622.

[J] B. J. Jiang, Estimation of the number of periodic orbits, Pacific J. Math. 172 (1996), 151–185.

[K] M. A. Kervaire, Le théorème de Barden-Mazur-Stallings, Comment. Math. Helv. 40 (1965), 31–42.

[M] J. Milnor, Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), Prindle, Weber & Schmidt, Boston, 1968, pp. 115–133.

[R] K. Reidemeister, Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1936), 586–593.

[RS] C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, Springer-Verlag, New York, 1972.

[St] J. Stallings, Centerless groups–An algebraic formulation of Gottlieb’s theorem, Topology 4 (1965), 129–134.

[Wa] C. T. C. Wall, Finiteness conditions for CW complexes, Ann. of Math. 81 (1965), 56–69.

[W] F. Wecken, Fixpunktklassen, I, II, III, Math. Ann. 117 (1941), 659–671; 118 (1942), 216–234; 118 (1942), 544–577.

Department of Mathematics, SUNY Binghamton, Binghamton, NY 13902–6000, USA E-mail address: ross@math.binghamton.edu

Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada E-mail address: nicas@mcmaster.ca