THE EULER CLASS OF A POINCARÉ DUALITY GROUP

IAN J. LEARY

Abstract. Under an extra hypothesis satisfied in every known case, we show that the Euler class of an orientable odd-dimensional Poincaré duality group over any ring has order at most two. We construct groups that are of type FL over the complex numbers but are not FL over the rationals. We construct group algebras over fields for which K_{0} contains torsion, and construct non-free stably-free modules for the group algebras of certain virtually-free groups.

1. Introduction

Let G be a group and R a (non-trivial, commutative, unital) ring. A number of finiteness conditions for the pair R and G have played a rôle in homological group theory and in algebraic topology. Classical examples include the conditions FP and FL. Both FP and FL are defined purely algebraically, although in the case when R = ℤ and G is finitely presented both have topological interpretations, discovered by C. T. C. Wall [26, 27]. Recently M. Bestvina and N. Brady introduced a topological condition, FH, at least as strong as FL, in the course of their work exhibiting non-finitely presented groups of type FL over ℤ [5]. (In fact, they show that their groups are FL by showing that they are FH.)

Roughly speaking, the difference between FL and FP is measured by an element E(G) in K_{0}(RG), which we shall call the Euler class of G. More precisely, if G is FP, then the image of E(G) in the reduced K-group K̃_{0}(RG) is zero if and only if G is FL. This image will be denoted Ẽ(G) and referred to as Wall’s finiteness obstruction. As general references we recommend [18, 23] for K_{0} and Wall’s finiteness obstruction, and [9] for finiteness conditions for groups.

Date: November 14, 2000.

1991 Mathematics Subject Classification. Primary 19A31; Secondary 16S34, 20J05, 55N15, 57P10.

Key words and phrases. Poincaré duality, finiteness conditions.

The author acknowledges support from the Nuffield Foundation, the CRM Barcelona, and from EPSRC grants GR/L06928, GR/L69398 and GR/R07813.

We introduce a new finiteness condition FPP, defined algebraically, and lying between FP and FL. (The extra ‘P ’ stands for ‘permutation’.) Just as every known torsion-free group of type FP is of type FL, it seems to be the case that every group of type FP is of type FPP. We show that the Euler class of an odd-dimensional orientable Poincaré duality group of type FPP has order at most two. In cases when K_{0} has no 2-torsion, it follows that such a group is necessarily of type FL. We exhibit groups G such that K_{0}(ℚG) contains torsion which is mapped to zero in K_{0}(ℂG). For some of these groups, we show that E(G) is a non-zero element of this kernel, and hence exhibit groups of type FP over ℚ that are FL over ℂ but not FL over ℚ. We deduce that over ℂ, FH is strictly stronger than FL.

Some of the techniques used for computing K_{0} also apply to the abelian monoid P of isomorphism types of finitely-generated projective modules. Hence we are able to give examples of non-free stably-free projective modules for certain group algebras.

Most of our calculations of K-groups (and the monoid P) rely on work of Waldhausen, or of Bergman and Dicks [2, 11, 25], but we do give a self-contained and comparatively elementary proof that there are groups that are FL over ℂ but not FL over ℚ. The definitions of all of the finiteness conditions mentioned above, and statements of our main results, are contained in the next section.

This work grew out of a study of the contents of [14]. Some examples with similar properties to ours were constructed using other methods in Boaz Moselle’s thesis [20]. The author thanks Alejandro Adem, Noel Brady, Ken A. Brown, Warren Dicks and Peter Kropholler for various helpful remarks concerning this work.

2. Definitions and Statements

For G and R as above, recall that G is said to be of type FP over R if there is a finite RG-projective resolution of R (viewed as an RGmodule by letting each g ∈ G act as the identity) in which each term is finitely generated, i.e., an exact sequence

0 → P_{m} → P_{m−1} → ⋯ → P_{1} → P_{0} → R → 0  (1)

in which each P_{i} is a finitely generated projective module [9], VIII.6. Similarly, G is said to be of type FL over R if there is an exact sequence as in (1) in which each P_{i} is a finitely generated free module.

We define G to be of type FPP over R if there exists a sequence as above in which each P_{i} is a finitely generated projective permutation module, i.e., a finite direct sum of modules of the form RG/H, where H is a finite subgroup of G whose order is a unit in R. Clearly, for any G

THE EULER CLASS OF A POINCARÉ DUALITY GROUP 3

and R, FL ⇒ FPP ⇒ FP, and if G is R-torsion-free, then FL ⇔ FPP. The author is not aware of any G and R such that FP ⇏ FPP. The existence of such G and R is a generalization of the question of whether there exists torsion-free G and R such that FP ⇏ FL. Examples of groups of type FP include any group G acting on an R-acyclic simplicial complex X with finitely many orbits of simplices, in such a way that the stabilizer of any simplex has order a unit in R. Every such G is FPP over R.

For any G of type FP over R, the Euler class E(G) is the element of K_{0}(RG) represented by the alternating sum

E(G) = ∑_{i=0}^{m}(−1)^{i}[P_{i}] ∈ K_{0}(RG),

where P_{i} is the ith module occurring in the projective resolution (1), and [P_{i}] stands for the corresponding class in K_{0}(RG). The Euler class depends only on (G, R) and not on the choice of resolution. It may be shown that E(G) is in the subgroup of K_{0}(RG) generated by [RG], the class of the free module, if and only if G is FL over R. Similarly, G is FPP over R if and only if E(G) is in the subgroup of K_{0}(RG) generated by elements of the form [RG/H].

G is said to be a Poincaré duality group of dimension n over R (or a PD^{n} group over R) if G is FP over R and as R-modules,

H^{i}(G, RG) ≅ \{ \begin{matrix}R & for i = n \\ 0 & for i ≠ n \end{matrix}

For any G, the RG-bimodule structure on RG gives rise to a right RGmodule structure on H^{∗}(G, RG). A PD^{n} group G over R is said to be orientable if each element of G acts as the identity on H^{n}(G, RG). If a group G acts on an R-acyclic simplicial complex X in the manner described above, and moreover, X is an R-homology n-manifold, then G is PD^{n} over R. (A simplicial complex is an R-homology n-manifold if the link of every i-simplex has the same R-homology as an (n−i−1)- sphere.)

Our first main result is the following:

Theorem 1. Let G be an orientable PD^{n} group over R for some odd n, and suppose that G is of type FPP over R. Then E(G) ∈ K_{0}(RG) has order at most two.

This theorem should be compared with the familiar fact that the Euler characteristic of a closed odd-dimensional manifold is zero. There are examples satisfying the hypotheses of the theorem for which E(G) is known to have order two. The first such example was a crystallographic group, discovered by P. H. Kropholler and B. Moselle, with the property that E(G) is an element of order two in K_{0}(kG) for any field of characteristic zero [14]. The proof of Theorem 1 uses a generalization of an argument presented in [14].

Theorem 1 leads naturally to the study of torsion in K_{0} of group algebras. One class of groups for which K_{0} can be explicitly computed is the class of virtually-free groups, to which work of G. Bergman, W. Dicks and F. Waldhausen applies [2, 11, 24, 25]. Using these results we are able to give examples as in the following statement, which will be proved in Section 6.

Theorem 2. For every integer n, and field k of characteristic zero, there is a virtually-free group G for which K_{0}(kG) contains an element of order n. For every prime p, there is a virtually-free group G for which K_{0}(ℚG) contains an element of order p, which is mapped to zero in K_{0}(ℂG).

The only published examples of torsion in K_{0}(kG) seem to be the examples due to Kropholler and Moselle [14], those contained in a recent paper of M. Lorenz [17], and our example in [15]. In [14, 17], crystallographic groups are given for which K_{0}(kG) contains 2-, 3-, and 4-torsion, for any field k of characteristic zero. In [15], a virtually-cyclic group G was exhibited, for which there is an element of order two in K_{0}(ℚG) mapping to zero in K_{0}(ℂG).

Of course, virtually-free groups are rarely Poincaré duality groups, but virtually-free quotients of a Poincaré duality group G can be used to detect elements of K_{0}(kG). Moreover, a relative version of Theorem 1 can be given (see Theorem 23 for an exact statement), which is applicable to some virtually-free groups. Using these ideas we prove the following statement, in Sections 5, 7 and 8.

Theorem 3. The following classes contain examples of groups that are FL over the complex numbers but not FL over the rationals:

(a) orientable crystallographic groups of dimension three;

(b) orientable PD^{3}-groups over the rationals containing non-cyclic free subgroups;

(c) virtually-free groups.

A ‘suspension lemma’, stated and proved as Lemma 20 in Section 7, allows one to deduce the following:

Corollary 4. For each n ≥ 3 there are PD^{n}-groups over the rationals that are FL over the complex numbers but not FL over the rationals.

THE EULER CLASS OF A POINCARÉ DUALITY GROUP 5

In both Theorems 2 and 3, more information concerning the behaviour under field extensions can be given, and will be given below. The simplest of the examples in Theorem 3 is the index two orientable subgroup of the group of reflections in the sides of a right regular triangular prism in ℝ^{3}. For this G, and any field k of characteristic not equal to 2 or 3, we compute K_{0}(kG) in Theorem 14. It transpires that G is of type FL if and only if k contains a primitive cube root of 1. The calculation for this example is modelled on the one in [14], but is complicated by the fact that we need to use Moody’s theorem [19] (as well as [2, 11, 24]).

In the proof of parts (a) and (b) of Theorem 3, the dependence on the work of Bergman, Dicks and Waldhausen can be reduced to showing that a certain projective module for a certain algebra (a quotient of each of the relevant group algebras) is not stably free. In [15] we gave an elementary proof of this fact using well-known facts about topological K-theory. In Section 10 we give a similar but more algebraic proof of this, by reducing to K_{0} of a Dedekind domain.

Groups having similar properties to the examples given in Theorems 2 and 3 were obtained by B. Moselle using different methods. Moselle’s thesis contained a detailed study of K_{0} of the rational group algebras of groups of the form IG: G, where G is finite and IG is the augmentation ideal in ℤG, viewed as a G-module [20]. Using a theorem of F. Quinn, announced in [22], Moselle exhibited groups of type FL over ℂ but not over ℚ, and groups for which E(G) ∈ K_{0}(ℚG) has order p for any prime p [20]. No full proof of Quinn’s theorem has appeared, and we consider it an advantage that our results do not depend upon it.

M. Bestvina and N. Brady say that G is of type FH over R if there is an R-acyclic simplicial complex upon which G acts freely with finitely many orbits [5]. For any G and R, FH ⇒ FL. In Section 9 we point out that FH is insensitive to field extensions, and hence that the groups listed in Theorem 3 cannot be of type FH over the complex numbers. In contrast, we show that over the rationals, FL ⇔ FH for groups having finite-type Eilenberg-Mac Lane spaces.

The techniques that we use to compute K_{0} for virtually-free groups also allow one to describe the abelian monoid of finitely-generated projective modules. In Section 11, we use these techniques to exhibit virtually-free groups G for which there are non-free stably-free kGmodules for any field k of characteristic zero.

3. Duality and Euler Classes

If M is an RG-module, the RG-bimodule structure on RG gives rise to a right RG-module structure on Hom_{RG}(M, RG). Define M^{∗} to be the RG-module obtained from N = Hom_{RG}(M, RG) by defining gn = ng^{−1} for any g ∈ G and n ∈ N. The assignment M ↦ M^{∗} is a contravariant functor. If M is a finitely generated projective module, then M^{∗∗} is naturally isomorphic to M.

Proposition 5. Let H be a finite subgroup of G. Then (RG/H)^{∗} ≅ RG/H.

Proof. It suffices to show that Hom_{RG}(RG/H, RG) ≅ RH\G, the right permutation module on the right cosets of H. But an element f in Hom_{RG}(RG/H, RG) is determined uniquely by f(1.H) = ∑_{g} r_{g}g, and this can be any element of RG fixed by the action of H, i.e., any element such that r_{g} = r_{hg} for all h ∈ H and g ∈ G. Let T be a right transversal to H in G. The set of elements of this form is right RG-isomorphic to RH\G via the map

∑_{g∈G} r_{g}g ↦ ∑_{t∈T} r_{t}Ht, with inverse ∑_{t∈T} r_{t}Ht ↦ ∑_{t∈T} ∑_{g∈Ht} r_{t}g.

Proof of Theorem 1. Since G is FPP over R, there is a resolution

0 → P_{m} → P_{m−1} → ⋯ → P_{1} → P_{0} → R → 0

for the trivial RG-module in which each P_{i} is a finitely generated projective permutation module. Necessarily, m ≥ n, since the cohomological dimension of G is equal to n, but equality need not hold. However, the image of the map from P_{n} to P_{n−1} is a finitely generated projective, Q_{n}, and there are long exact sequences

0 → P_{m} → P_{m−1} → ⋯ → P_{n} → Q_{n} → 0,  (2)
0 → Q_{n} → P_{n−1} → ⋯ → P_{1} → P_{0} → R → 0.  (3)

Each term in sequence (2) is projective, so the sequence splits. It follows that the sequence obtained by applying the functor (−)^{∗} is also exact:

0 → Q_{n}^{∗} → P_{n}^{∗} → ⋯ → P_{m}^{∗} → 0.

By Proposition 5, P_{i} ≅ P_{i}^{∗} for each i. It follows that in K_{0}(RG),

[Q_{n}^{∗}] = ∑_{i=0}^{n−m}(−1)^{i}[P_{n−i}^{∗}] = ∑_{i=0}^{n−m}(−1)^{i}[P_{n−i}] = [Q_{n}].

Now apply (−)^{∗} to the projective resolution of R over RG given by sequence (3). Since G is an orientable Poincaré duality group, this gives rise to an exact sequence

0 → P_{0}^{∗} → P_{1}^{∗} → ⋯ → P_{n−1}^{∗} → Q_{n}^{∗} → R → 0.  (4)

Sequence (3) gives the expression

E(G) = (−1)^{n}[Q_{n}] + ∑_{i=0}^{n−1}(−1)^{i}[P_{i}]

for E(G). On the other hand, sequence (4) gives

E(G) = [Q_{n}^{∗}] + ∑_{j=1}^{n}(−1)^{j}[P_{n−j}^{∗}] = [Q_{n}] + ∑_{i=0}^{n−1}(−1)^{n−i}[P_{i}].

Since n is odd, it follows that E(G) = −E(G) as required.

Theorem 1, and its proof, are closely related to theorem 1.3 of C. T. C. Wall’s paper [28]. Wall mentions only the case R = ℤ, but the generalization to any R is immediate. On the other hand, Wall does not impose any condition corresponding to our FPP, so obtains a weaker conclusion.

Under certain circumstances Theorem 1 may be used to show that a Poincaré duality group is of type FL.

Corollary 6. Let G contain only finitely many conjugacy classes of finite subgroups, let k be a splitting field for every finite subgroup of G, and suppose that G contains no element of order two. If G is an orientable PD^{n} group over k of type FPP for some odd n, then G is FL.

Proof. For any G, let Φ(G) be the Frobenius category of G. The objects of Φ are the finite subgroups of G, and the morphisms from H to H′ are those group homomorphisms from H to H′ that are expressible as conjugation by some element of G. For any k, the natural maps K_{0}(kH) → K_{0}(kG) induce a map

ι: colim_{H∈Φ} K_{0}(kH) → K_{0}(kG).

If G is of type FPP over k, the Euler class of G is in the image of ι.

Now suppose that G and k satisfy the hypotheses of the first sentence of the statement. In this case, work of K. A. Brown and M. Lorenz shows that colim_{H∈Φ} K_{0}(kH) is a finitely generated abelian group containing no 2-torsion ([8], 2.5). An argument using Hattori-Stallings ranks (implicit in [8], section 1) shows that the kernel of ι is finite. It follows that there is no 2-torsion in the image of ι.

Now suppose that G and k also satisfy the hypotheses of the second sentence of the statement. By Theorem 1, E(G) has order at most two, but by the above argument E(G) cannot have order two, and so G is FL.

4. Computational Techniques

In this section we summarise the techniques for computing K_{0}(RG) that will be used in the sequel. For the definition of a graph of groups, see [12]. The following theorem is from [24], but see also [2, 11, 25].

Theorem 7 (Waldhausen). Let G be the fundamental group of a finite graph of groups, with vertex groups G_{1}, …, G_{m} and edge groups H_{1}, …, H_{n}. Let k be a field and suppose that each kG_{i} is regular coherent and that each kH_{j} is regular Noetherian. There is an exact sequence

⨁_{j=1}^{n} K_{0}(kH_{j})⟶^{ι} ⨁_{i=1}^{m} K_{0}(kG_{i}) → K_{0}(kG) → 0,

where the map ι is induced by the inclusion of each edge group in its initial vertex group minus the inclusion in its terminal vertex group.

Remarks on the proof. The conditions on kG_{i} and kH_{j} hold if each G_{i} is polycyclic-by-finite and the order of every finite subgroup of every G_{i} is invertible in k. (This is the only case in which we shall use the Theorem.) If these conditions hold, it may be shown that kG is also regular coherent. Hence the Theorem may be proved by induction on the size of the graph, using sections 11 and 12 of [25], which cover the cases of a free product with amalgamation and an HNN-extension respectively. The result required for exactness at K_{0}(kG) is not explicitly stated in section 12, but works in the same way as similar results stated in sections 11 and 13 of [25].

In the case when each H_{i} is finite, which will serve for all of our calculations except those in Section 5, Theorem 7 may be deduced from work of G. Bergman and W. Dicks, see in particular corollary 2.11 of [2] and theorem 26 of [11], which cover the case of a free product with amalgamation and an HNN-extension respectively. The alternative approach of Bergman-Dicks also describes the abelian monoid of finitely-generated projective kG-modules. This will be used in Section 11 to exhibit non-free stably-free modules. A shorter statement of Theorem 7 is the following: Make a category C whose objects are the vertex and edge groups, with two non-identity morphisms from each edge group, the homomorphisms to its initial and terminal vertex groups. Then the following map is an isomorphism.

colim_{H∈C} K_{0}(kH) → K_{0}(kG)

The graphs of groups used below are of a particularly simple type. These graphs are described by a group J and a family H_{1}, …, H_{n} of

THE EULER CLASS OF A POINCARÉ DUALITY GROUP 9

(not necessarily distinct) subgroups of J. There are two vertex groups, J_{1} and J_{2}, identified with J by some fixed isomorphism, and edge groups H_{1}, …, H_{n}. Each edge joins the two vertices, and the map from H_{i} to J_{j} is the inclusion of H_{i} in J. Call the fundamental group of such a graph the double of J along H_{1}, …, H_{n}.

Proposition 8. Let G be the double of a polycyclic-by-finite group J along H_{1}, …, H_{n}. Let k be a field in which the order of every finite subgroup of J is invertible. Then there are isomorphisms

K_{0}(kG) ≅ K_{0}(kJ) ⊕ coker (ι: ⨁_{i=1}^{n} K_{0}(kH_{i}) → K_{0}(kJ)) ,

and

K̃_{0}(kG) ≅ K̃_{0}(kJ) ⊕ coker (ι: ⨁_{i=1}^{n} K_{0}(kH_{i}) → K_{0}(kJ)) .

Under these hypotheses G is of type FP over k, and the above isomorphism sends E(G) to the element

(2E(J) − ∑_{i=1}^{n} Ind_{H_{i}}^{J}E(H_{i}), E(J)).

Proof. Let A = K_{0}(kJ). By Theorem 7, K_{0}(kG) is isomorphic to the quotient (A ⊕ A)/{(ι(x), −ι(x))}, where x ∈ ⨁ K_{0}(kH_{i}). The automorphism of A⊕A given by (a, a′) ↦ (a+a′, a′) induces an isomorphism between this and (A ⊕ A)/{(0, −ι(x))} = A ⊕ coker(ι).

The augmented chain complex for the G-tree coming from the graph product decomposition is the following exact sequence:

0 → ⨁_{i=1}^{n} kG/H_{i} → kG/J_{1} ⊕ kG/J_{2} → k → 0.

Any subgroup of a polycyclic-by-finite group is polycyclic-by-finite, and such a group admits a proper cocompact action on a contractible manifold [22]. It follows that J and the H_{i} are all of type FP over k, and hence so is G. Moreover, it follows that the Euler class of J is the sum of the images of E(J_{1}) and E(J_{2}) minus the sum of the images of the E(H_{i}). Tracing this element through the isomorphism given in the first paragraph gives the claimed element of A ⊕ coker(ι).

It is possible to prove some results concerning doubles without using Theorem 7. In the following proposition we need no condition on J except that it be FP over R.

Proposition 9. Let H ≤ J be groups of type FP over R. Let G be the double of J along H, so that G is FP over R. If E(J) ∈ K_{0}(RJ) is in the image of Ind_{H}^{J} and Ind_{H}^{J}(E(H)) = 2E(J), then E(G) = 0.

Proof. As in Proposition 8, consideration of the chain complex for the G-tree coming from the given splitting shows that

E(G) = Ind_{J_{1}}^{G}(E(J_{1})) + Ind_{J_{2}}^{G}(E(J_{2})) − Ind_{H}^{G}(E(H)),

where J_{1} and J_{2} are the two copies of J in G. If E(J) = Ind_{H}^{J}(ξ), then

E(G) = Ind_{J_{1}}^{G}(E(J_{1})) + Ind_{J_{2}}^{G}Ind_{H}^{J_{2}}(ξ) − Ind_{H}^{G}(E(H))
= Ind_{J_{1}}^{G}(E(J_{1})) + Ind_{H}^{G}(ξ − E(H))
= Ind_{J_{1}}^{G}(E(J_{1}) + Ind_{H}^{J_{1}}(ξ − E(H)))
= Ind_{J_{1}}^{G}(2E(J_{1}) − Ind_{H}^{J_{1}}(E(H)))

There is an argument similar to that used in the proof of Theorem 1 to exhibit some cases in which Ind_{H}^{J}(E(H)) = 2E(J). For a Poincaré duality group G over R, the orientation module, R^{∘}, is the left RGmodule corresponding to the right module H^{∗}(G; RG). In every known case, the action of G on its orientation module has image contained in {±1} ⊆ Aut_{R}(R). We shall say that G is standard if this holds. The question of whether every Poincaré duality group is standard is posed by M. Bestvina in section 2.2 of [4]. Note that the topological definition of a Poincaré duality group used in [4] is far more restrictive than the algebraic definition used in this paper, which is equivalent to the definition of R. Bieri and B. Eckmann [6, 7, 9].

Lemma 10. Let G be a standard, non-orientable PD^{n} group over R for some even n, and let H be the kernel of the orientation map. If G is of type FPP over R, then Ind_{H}^{G}(E(H)) = 2E(G).

Proof. Take a finite projective resolution for R over RH, and apply the exact functor RG ⊗_{RH} −. Inspection of this long exact sequence shows that Ind_{H}^{G}(E(H)) is equal to the alternating sum of the modules in a finite projective resolution for RG/H. By the hypotheses on G, there is a short exact sequence of RG-modules:

0 → R^{∘} → RG/H → R → 0.

It follows that Ind_{H}^{G}(E(H)) = 2E(G) if and only if the alternating sum of the modules in a finite projective resolution for R^{∘} equals E(G). This can be proved using the same trick as in the proof of Theorem 1. There is a projective resolution for R over RG of the form

0 → Q_{n} → P_{n−1} → ⋯ → P_{1} → P_{0} → R → 0,  (5)

THE EULER CLASS OF A POINCARÉ DUALITY GROUP 11

where each P_{i} is a projective permutation module, and Q_{n} is such that [Q_{n}^{∗}] = [Q_{n}] in K_{0}(RG). Applying the functor (−)^{∗} to this projective resolution gives rise to a long exact sequence

0 → P_{0}^{∗} → P_{1}^{∗} → ⋯ → P_{n−1}^{∗} → Q_{n}^{∗} → R^{∘} → 0,  (6).

The claim follows by comparing sequences (5) and (6).

A polycyclic-by-finite group G admits a proper, cocompact action on a contractible manifold (see for example the introduction to [22]). It follows that G is of type FP over R if and only the order of every finite subgroup of G is a unit in R. Moreover, in this case, G is of type FPP and is a Poincaré duality group over R, of dimension equal to that of the manifold. It also follows that G is standard in the sense of Lemma 10.

Corollary 11. Let J be polycyclic-by-finite, and let k be a field over which J is a non-orientable PD^{2n} group. Let H be the orientable subgroup of J, and let G be the double of J along H. Then G is an orientable PD^{2n+1} group over k, and E(G) ∈ K_{0}(kG) has order two if and only if E(J) is not in the image of ι: K_{0}(kH) → K_{0}(kJ).

Proof. An easy spectral sequence argument shows that the direct product of a PD^{m} group and a PD^{l} group is a PD^{m+l} group, with orientation module the tensor product (over k) of the orientation modules for the two factors. The group G may be checked to be the orientable index two subgroup of the direct product of J and an infinite dihedral group, which proves the first claim. The second claim follows from Proposition 8 and Lemma 10.

To apply the above, a method for calculating K_{0} for some polycyclicby-finite groups is needed. The following theorem is from [19].

Theorem 12 (Moody). Let G be polycyclic-by-finite, and let Φ(G) be the Frobenius category of G, as described in the proof of Corollary 6. For any Noetherian R, the natural map

ι: colim_{H∈Φ} K_{0}(RH) → K_{0}(RG)

is a surjection.

Quinn’s theorem [22] states that in the case when R = ℚ, the map ι as above is an isomorphism. We shall not need to use this however.

5. The Prism Group

In this section we study a single example, namely the group G defined to be the orientable subgroup of the group generated by reflections in the sides of a right regular triangular prism in ℝ^{3}. For this G, and for any field k of characteristic not equal to 2 or 3, we compute K_{0}(kG), and show that E(G) ∈ K_{0}(kG) has order two if and only if k does not contain a primitive cube root of unity.

Let J be the (3, 3, 3)-triangle group, i.e., the group generated by the reflections in the sides of a regular triangle in the Euclidean plane. The group J has the presentation

J = ⟨a, b, c | a^{2}, b^{2}, c^{2}, (ab)^{3}, (bc)^{3}, (ca)^{3}⟩.

The group G is the orientable subgroup of the product of J and the infinite dihedral group, and may also be viewed as the double of J along H, its orientable subgroup of index two. Before starting our calculations, we record some facts about J and H.

The group J is expressible as an extension with kernel ℤ^{2} and quotient the dihedral group D_{3} of order six. J has three conjugacy classes of maximal finite subgroup, each isomorphic to D_{3}. All elements of J of order two are conjugate. Every element of J of order three is conjugate to its inverse, and there are three conjugacy classes of subgroup of J of order three, one contained in each conjugacy class of D_{3}. The group H has three conjugacy classes of maximal finite subgroup, each cyclic of order three, and has six conjugacy classes of elements of order three. All of these facts may be verified easily by considering the action of J on the Euclidean plane, bearing in mind that any finite subgroup K of J must have a fixed point (for example, the centroid of any K-orbit).

Lemma 13. Let J, H and k be as above, and let ω be a primitive cube root of unity in some extension of k. Then K_{0}(kJ) ≅ ℤ^{⊕5},

coker(Ind_{H}^{J}) = \{ \begin{matrix}ℤ & if ω ∈ k, \\ ℤ ⊕ (ℤ_{2})^{⊕3} & otherwise, \end{matrix}

and E(J) is in the image of Ind_{H}^{J} if and only if ω ∈ k.

Proof. First we compute the limit of K_{0}(k−) over the Frobenius category of J. Since J is abelian-by-finite, and all maximal finite subgroups of J map isomorphically to a fixed finite quotient of J, this limit is easy to describe. It is isomorphic to the quotient of three copies of K_{0}(kD_{3}) in which the three images of Ind_{C_{2}}^{D_{3}} have been identified. (Here C_{2} denotes a subgroup of D_{3} of order two.) Since the characteristic of k is not equal to two or three, there are three irreducible kD_{3}-modules, the trivial module (denoted ‘1’), a non-trivial 1-dimensional module ϵ, and a 2-dimensional module α. The image of the induction from C_{2} is generated by 1 + α and ϵ + α. Listing the three kD_{3}-modules in the order 1, ϵ, α for i = 1, 2, 3, the limit over the Frobenius category is ℤ^{⊕9} modulo the subgroup generated by the rows of matrix (7), in which blank entries are zeroes omitted for clarity.

\begin{aligned}1 & 0 & 1 & & & & −1 & 0 & −1 \\ 0 & 1 & 1 & & & & 0 & −1 & −1 \\ & & & 1 & 0 & 1 & −1 & 0 & −1 \\ & & & 0 & 1 & 1 & 0 & −1 & −1 \end{aligned}  (7)

This is isomorphic to ℤ^{⊕5} as claimed.

By Moody’s theorem, quoted as Theorem 12, it follows that K_{0}(kJ) is a quotient of ℤ^{⊕5}. But J contains five conjugacy classes of elements of finite order, and Hattori-Stallings traces on these elements (or their Brauer equivalent if the characteristic of k is non-zero) suffice to show that the map from ℤ^{⊕5} to K_{0}(kJ) is injective, and hence is an isomorphism.

Let C_{3} denote a cyclic group of order three. If ω ∉ k, there are two irreducible kC_{3}-modules, the trivial module and a 2-dimensional module. The images of these under Ind_{C_{3}}^{D_{3}} are 1+ϵ and 2α respectively. On the other hand, if ω ∈ k, there are three irreducible kC_{3}-modules, each of dimension one, and their images under Ind_{C_{3}}^{D_{3}} are 1 + ϵ, α, and α respectively.

Applying Moody’s theorem to H, the image of Ind_{H}^{J} is equal to the sum of the images of Ind_{K}^{J}, where K ranges over the maximal finite subgroups of H. Representing K_{0}(kJ) as the quotient of ℤ^{⊕9} by the rows of (7), the image of Ind_{H}^{J} is generated by

\begin{aligned}1 & 1 & 0 \\ 0 & 0 & 2 & & & & & & & ∗ \\ & & & 1 & 1 & 0 \\ & & & 0 & 0 & 2 & & & & ∗ \\ & & & & & & 1 & 1 & 0 \\ & & & & & & 0 & 0 & 2 & ∗ \end{aligned}  (8)

if ω ∉ k. In the case when ω ∈ k the matrix (8′) representing the image of Ind_{H}^{J} is similar to (8), except that the three rows of (8) marked with an asterisk are divided by two. The cokernel of Ind_{H}^{J} is isomorphic to the quotient of ℤ^{⊕9} by the subgroup generated by the rows of (7) and (8) if ω ∉ k (resp. (7) and (8′) if ω ∈ k). The claimed description of this cokernel follows.

The Euler class of J may be computed by considering the chain complex for the J-equivariant tesselation of the plane by triangles. The triangular 2-cells are permuted freely, transitively. There are 3 orbits of 1-cell, each with stabilizer of order two, and there are three orbits of 0-cell, with stabilizers the three conjugacy classes of D_{3} subgroup.

Hence the Euler class of J is represented by the vector

(1, 1, 2, 0, 0, 0, 0, 0, 0)−3.(1, 0, 1, 0, 0, 0, 0, 0, 0)+(1, 0, 0, 1, 0, 0, 1, 0, 0)

= (−1, 1, −1, 1, 0, 0, 1, 0, 0).

It may be checked that this vector is in the subgroup generated by the rows of (7) and (8′) but not in the subgroup generated by the rows of (7) and (8).

Theorem 14. Let J, H, and k be as in the introduction to the section, and let ω be a primitive cube root of unity in some extension of k. Let l be the degree of k[ω] over k, and let G be the double of J along H. Then G is an orientable PD^{3} group over k of type FPP,

K_{0}(kG) ≅ ℤ^{⊕6} ⊕ (ℤ_{l})^{⊕3},

and E(G) ∈ K_{0}(kG) has order l. In particular, G is FL over k if and only if l = 1.

Proof. This follows from Corollary 11 and Lemma 13.

One can also obtain some information for integral domains. Suppose R is any integral domain in which 6 is a unit. Then E(G) ∈ K_{0}(RG) is defined and has order at most two by Theorem 1. If R contains a primitive cube root of unity it may be shown that E(G) = 0. On the other hand, if the field of fractions of R does not contain a primitive cube root of unity, the order of E(G) is two.

6. Virtually-Free Groups

In this section we show how computations in the K-theory of coproducts can be used to construct virtually-free groups with torsion in their group algebras. Perhaps the most surprising examples are groups G and field extensions K ≥ k such that the map from K_{0}(kG) to K_{0}(KG) has non-trivial kernel. Simpler such examples can be given that are not group algebras, so we discuss these first.

Suppose that K is an extension field of k of degree n. The action of K on itself gives a k-algebra embedding of K in the matrix ring S = M_{n}(k). Now K_{0}(K) is isomorphic to ℤ, with generator the free module of rank 1, and K_{0}(S) is isomorphic to ℤ, generated by the module of column vectors, k^{n}, so that the free module represents n times a generator. Let R be the coproduct of two copies of S, amalgamating K. The results of Bergman or Waldhausen, stated in a special case as Theorem 7, give an exact sequence

K_{0}(K) → K_{0}(S) ⊕ K_{0}(S) → K_{0}(R) → {0}.

THE EULER CLASS OF A POINCARÉ DUALITY GROUP 15

It follows that K_{0}(R) is isomorphic to ℤ ⊕ ℤ_{n}. On the other hand, if \overline{k} is the algebraic closure of k, then \overline{k} ⊗_{k} K ≅ \overline{k} × ⋯ × \overline{k} = \overline{k}^{n}, \overline{k} ⊗_{k} S ≅ M_{n}(\overline{k}), and the image of any of the n simple projective modules for \overline{k}^{n} maps to a generator for K_{0}(M_{n}(\overline{k})). Hence K_{0}(\overline{k} ⊗_{k} R) is isomorphic to ℤ, generated by the free module.

The ring R is not a group algebra, but there is a case in which R can be viewed as a skew group algebra: if K is a Galois extension of k, with Galois group G, then the natural skew group algebra K ∗ G is isomorphic to M_{n}(k) (see [10], p.590). Hence R may be viewed as the skew group algebra of the free product G ∗ G with coefficients K, where each factor acts in the natural way on K. In this case \overline{k} ⊗_{k} R is the skew group algebra with coefficients \overline{k}^{n}, where each of the two copies of G acts by freely permuting the n copies of \overline{k}. The special case of this construction in which k = ℝ and K = ℂ will be discussed in detail in Section 10. The group algebras given below in Propositions 15 and 16 are modelled on these examples.

Proposition 15. Let p and q be distinct primes, with q dividing p−1, and let J be a non-abelian group of order pq. Let G be the double of J along H, the order p subgroup of G. Then K_{0}(ℚG) contains an element of order q, which maps to zero in K_{0}(ℂG).

Proof. The group J has three irreducible rational representations: the trivial representation T of dimension 1, a (q − 1)-dimensional representation α, faithful on the order q quotient of J, and a (p−1)-dimensional representation β. The regular representation of J is equal to T +α+qβ. The image of induction from H is generated by T + α and the regular representation, and so the cokernel of this map contains q-torsion. From Proposition 8 it follows that K_{0}(ℚG) contains q-torsion. In fact, K_{0}(ℚG) ≅ ℤ^{4} ⊕ ℤ_{q}.

On the other hand, J has q distinct 1-dimensional irreducible complex representations, each containing H in its kernel, and (p−1)/q distinct q-dimensional irreducible complex representations, each of which is induced up from H. It follows that the image of K_{0}(ℂH) in K_{0}(ℂJ) is a direct summand, and hence K_{0}(ℂG) is free abelian (of rank 2q − 1 + (p − 1)/q).

We give more information in the special case of dihedral groups.

Proposition 16. Let p be an odd prime, and let G be the double of the dihedral group of order 2p along its cyclic subgroup of order p. Let k be a field of characteristic zero, and let ζ be a primitive pth root of 1 in some extension of k. Finally, define m to be the degree of the field extension m = |k[ζ + ζ^{−1}] : k|, and let r = (p − 1)/2m. Then

K_{0}(kG) ≅ \{ \begin{matrix}ℤ^{⊕r+3} & if ζ ∈ k[ζ + ζ^{−1}], \\ ℤ^{⊕r+3} ⊕ (ℤ_{2})^{⊕r} & otherwise. \end{matrix}

Proof. Define l by the equation l = |k[ζ] : k|. The irreducible krepresentations of C_{p} are the trivial representation of dimension one, and (p − 1)/l others of dimension l. The irreducible k-representations of D_{p} are the trivial representation, the orientation representation of dimension one, and r = (p − 1)/2m others of dimension 2m. In the case when l = m, or equivalently when ζ ∈ k[ζ + ζ^{−1}], the image of induction from C_{p} to D_{p} is spanned by the sum of the two 1-dimensional representations and the other representations. Hence in this case the cokernel of induction is infinite cyclic. In the case when l = 2m, each non-trivial irreducible representation of C_{p} induces up to two copies of one of the irreducible representations of D_{p}, and the cokernel of induction is infinite cyclic direct sum r copies of ℤ_{2}. The claims follow from Proposition 8.

Proposition 17. Let p be a prime and let J be a direct product of m+1 cyclic groups of order p. Let H_{1}, …, H_{n} be the order p subgroups of J, so that n = (p^{m+1} − 1)/(p − 1). Let G be the double of J along H_{1}, …, H_{n}. For any field k of characteristic not equal to p, K_{0}(kG) contains elements of order p^{m}.

Proof. For i = 1, …, n, let T_{i} (resp. R_{i}) denote the trivial (resp. regular) representation of H_{i}, and similarly let T_{0} (resp. R_{0}) denote the trivial (resp. regular) representation of J. By hypothesis each T_{i} is projective. By symmetry, there are integers r and s such that

ι (∑_{i=1}^{n} T_{i}) = rT_{0} + sR_{0}.

A comparison of k-dimensions and numbers of trivial summands shows that r = p^{m} and s = (p^{m} − 1)/(p − 1). Since ι(R_{1}) = R_{0}, it follows that p^{m}T_{0} is in the image of ι.

On the other hand, the index of each H_{j} in J is p^{m}, so any kJmodule in the image of ι has dimension divisible by p^{m}. It follows that the subgroup of coker(ι) generated by T_{0} is cyclic of order p^{m}.

Corollary 18. For any positive integer n and any field k of characteristic coprime to n, there is a virtually-free group G such that K_{0}(kG) contains elements of order n.

Proof. Take the free product of examples as in Proposition 17 for the distinct prime powers dividing n.

THE EULER CLASS OF A POINCARÉ DUALITY GROUP 17

These examples answer a question posed in section 2.5 of [8], as do certain examples in Moselle’s thesis [20]. Lemma 2.2 of [8] shows that our methods will never find p-torsion in K_{0}(kG) for k of characteristic p.

7. Poincaré Duality Groups

In this section we combine results of previous sections to exhibit more examples of Poincaré duality groups over the rationals that are FL over the complex numbers but not FL over the rationals. Our first such example is the index-two orientable subgroup of the direct product of a (p, p, p)-triangle group and an infinite dihedral group, for p ≥ 5 a prime. The calculations are very similar to those in Section 5, except that we cannot fully determine K_{0}, since no analogue of Moody’s theorem (Theorem 12) is known.

Higher dimensional examples could be constructed using similar techniques. It is, however, easier to apply the ‘suspension lemma’, given below as Lemma 20, to obtain higher dimensional examples from those constructed already.

Let J be the (p, p, p)-triangle group, that is, the group generated by reflections in the sides of a hyperbolic triangle with all angles 2π/p. Arguments similar to those outlined in Section 5 show that J has three conjugacy classes of maximal finite subgroup, each dihedral of order 2p, three conjugacy classes of cyclic subgroup of order p (one in each class of dihedral group) and one conjugacy class of elements of order two. Let D(i) and C(i) for i = 1, 2, 3 be representatives of the three conjugacy classes of D_{p} subgroups and C_{p} subgroups respectively. There is a homomorphism π from J to D_{p} that restricts to an isomorphism on each D(i), and hence has torsion-free kernel. (Just send the three generators of J of order two to any three distinct elements of D_{p} of order two.) The kernel of any such π is the fundamental group of a closed orientable surface of Euler characteristic 3−p. The image of the orientable index-two subgroup H of J under π is C_{p} ≤ D_{p}.

Theorem 19. Let k be a field of characteristic zero, let ζ be a primitive pth root of 1 in some extension of k, let H, J be as described above, and let G be the double of J along H. Then G is an orientable PD^{3}-group over k, and E(G) ∈ K_{0}(kG) has order two if ζ ∉ k[ζ + ζ^{−1}], and is equal to zero otherwise.

Proof. The group G is the orientable index-two subgroup of J × D_{∞}, so is an orientable PD^{3}-group of type FPP over k. It follows from Theorem 1 that E(G) has order at most two.

Let ϵ denote the non-trivial 1-dimensional representation of D_{p}, and define a representation α of D_{p} by the equation kD_{p} = 1+ϵ+2α. Other equivalent descriptions of α are that α consists of a direct sum of one copy of each irreducible k-representation of D_{p} of dimension greater than one, or that kD_{p}/C_{2} = 1 + α. Arguments similar to those used in the proof of Proposition 16 show that α is in the image of induction from kC_{p} if and only if ζ ∈ k[ζ + ζ^{−1}].

The natural J-equivariant triangulation of the hyperbolic plane has one free orbit of 2-simplices, three orbits of edges with stabilizer of order two, and three orbits of vertices with stabilizers conjugate to D(1), D(2), and D(3) respectively. It follows that the Euler class of J is

E(J) = [kJ] − 3[kJ/C_{2}] + ∑_{i=1}^{3}[kJ/D(i)]

= [kJ] − ∑_{i=1}^{3}[Ind_{D(i)}^{J}(α)].

If ζ ∈ k[ζ+ζ^{−1}], there is a k-representation ξ of C_{p} such that Ind_{C_{p}}^{D_{p}}(ξ) = α. Hence E(J) is in the image of induction from H, because

E(J) = Ind_{H}^{J} ([kH] − ∑_{i=1}^{3}[Ind_{C(i)}^{H}(ξ)) .

By Proposition 9 and Lemma 10, it follows that in this case E(G) = 0.

For the converse, note that the image of E(J) under the homomorphism π : J → D_{p} is

[kD_{p}] − 3α = 1 + ϵ − α.

If ζ ∉ k[ζ + ζ^{−1}], then this is a non-zero element in the cokernel of the induction from C_{p} to D_{p}. From the description of E(G) and K_{0}(kG) given in Proposition 8, it follows that in this case E(G) is non-zero.

The easiest way to exhibit higher-dimensional Poincaré groups that are FL over the complex numbers but not FL over the rationals is to apply the following.

Lemma 20. Let R be a ring in which 2 is a unit, let G be a group of type FP over R, and let D be an infinite dihedral group. Then E(G) = 0 in K_{0}(RG) if and only if E(G × D) = 0 in K_{0}(RG × D).

Proof. Let a and b denote generators of order two for D, and let Q_{∗} denote the RD-projective resolution of R coming from the chain complex for the line on which D acts with one edge orbit and two vertex orbits:

Q_{∗} = 0 → RD → RD/⟨a⟩ ⊕ RD/⟨b⟩ → R → 0.

THE EULER CLASS OF A POINCARÉ DUALITY GROUP 19

Now let P_{∗} be any finite RG-projective resolution for R. The total complex of P_{∗} ⊗ Q_{∗}, where the tensor product is over R, is a finite RG × D-projective resolution for R. If E(G) = 0, then there exists a finitely generated projective P such that

P ⊕ ⨁_{i} P_{2i} ≅ P ⊕ ⨁_{i} P_{2i+1}.

Tensoring this equation over R with Q_{0}, and with Q_{1} and adding gives

P ⊗ (Q_{0} ⊕ Q_{1}) ⊕ ⨁_{i} P_{2i} ⊗ Q_{0} ⊕ ⨁_{i} P_{2i+1} ⊗ Q_{1} ≅

P ⊗ (Q_{0} ⊕ Q_{1}) ⊕ ⨁_{i} P_{2i+1} ⊗ Q_{0} ⊕ ⨁_{i} P_{2i} ⊗ Q_{1},

and hence E(G × D) = 0.

For the converse, note that the map from K_{0}(RD) to K_{0}(R) induced by the unique homomorphism from D to the trivial group sends Q_{0} to the free module of rank two, and sends Q_{1} to the free module of rank one. It follows that the homomorphism induced by the projection from G × D onto G with kernel D sends

E(G × D) = ∑_{i,j}(−1)^{i+j}[P_{i} ⊗ Q_{j}]

to

2. ∑_{i}(−1)^{i}[P_{i}] + ∑_{i}(−1)^{i+1}[P_{i}] = E(G).

Hence if E(G × D) = 0, then E(G) = 0. Proof of Corollary 4. In the case when n = 3, PD^{n}-groups over ℚ that are FL over ℂ but not FL over ℚ were exhibited in Theorems 2, 14 and 19. If G is an example of an n-dimensional group having these properties, Lemma 20 implies that the product of G and an infinite dihedral group is an example of dimension n + 1.

Other groups may be used instead of the infinite dihedral group in Lemma 20. The following Lemma may be proved in the same way.

Lemma 21. Let H be either the free group of rank two, or the fundamental group of a closed surface of Euler characteristic −1. Let G be of type FP over R. Then E(G) = 0 if and only if E(G × H) = 0, and G is FL over R if and only if G × H is FL over R.

Compared to the infinite dihedral group, these two groups have the advantages that they are torsion-free, so no restriction on R is needed, and that they are FL over any R. (Since the dihedral group D is not FL over any (non-trivial) ring, it can happen that G is FL over R, but G×D is not.) On the other hand, the product of G and the free group of rank two cannot be a Poincaré duality group. The fundamental group, H, of a surface of Euler characteristic −1 does not have this defect, but of course taking a direct product with H increases cohomological dimension by two. Both Lemma 20 and Lemma 21 are special cases of a product formula for Euler classes described in [18].

8. More Virtually-Free Groups

The Euler class of a virtually-free group can never be a non-zero torsion element, as shown below in Proposition 22. It can, however, happen that the Wall finiteness obstruction Ẽ(G) for a virtually-free group G is a torsion element in K̃_{0}(RG), the quotient of K_{0}(RG) by the subgroup generated by [RG]. One example of this appeared (unremarked) earlier in the paper: the case p = n = 2 of Proposition 17. In this section, such examples are exhibited using Theorem 23, a relative version of Theorem 1.

Proposition 22. Let G be a (finitely generated) virtually-free group and R a ring over which G is FP. Then E(G) ∈ K_{0}(RG) is either zero or has infinite order.

Proof. The Euler class of a non-cyclic free group has infinite order. It follows that if G has a non-cyclic free subgroup, then E(G) has infinite order, and it remains to consider the virtually-cyclic case. If G has an infinite cyclic subgroup of finite index, then either G = N: ℤ, or G = A ∗_{N} B, where N is a finite normal subgroup, and in the second case |A : N| = |B : N| = 2 ([12], IV.6.12). These two cases correspond to actions on a line with one orbit of edges and one or two orbits of vertices respectively. In the first case the edge and vertex stabilizers are both equal to N, so E(G) = [RG/N] − [RG/N] = 0. In the second case, E(G) = [RG/A] + [RG/B] − [RG/N], and has infinite order, since the Hattori-Stallings trace of E(G) on any element of A − N is non-zero.

Theorem 23. Let X be a contractible triangulated n-manifold with boundary for some odd n, and suppose that G acts on X, preserving the orientation of X, with finitely many orbits of simplices, freely on the boundary of X, and with stabilizers of order invertible in R. Then G is FPP over R, and the Wall obstruction Ẽ(G) ∈ K̃_{0}(RG) has order at most two.

Proof. Replacing X by its barycentric subdivision, we may suppose that the simplicial chain complex C_{∗} = C_{∗}(X; R) (with R-basis the simplices of X) is a sum of permutation modules RG/H_{j} where each |H_{j}| is a unit in R. Since X is contractible, C_{∗} is an RG-projective resolution for R, G is FPP over R, and

E(G) = ∑_{i}(−1)^{i}[C_{i}] ∈ K_{0}(RG).  (9)

Let D^{∗} be the cochain complex with compact supports for X relative to its boundary δX, so that D^{∗} has R-basis the characteristic functions of those simplices of X that are not contained in δX. Since G acts freely on δX, for each i, there is a finitely generated free RG-module F_{i} such that D^{i} ⊕ F_{i} ≅ C_{i}. By Poincaré duality for X (see [21]), there is a long exact sequence

0 → D^{0} → D^{1} → ⋯ → D^{n−1} → D^{n} → R → 0.

It follows that

E(G) = ∑_{i}(−1)^{n−i}[D^{i}] =

(−1)^{n} ∑_{i}(−1)^{i}[C_{i}] + (−1)^{n+1} ∑_{i}(−1)^{i}[F_{i}].  (10)

Since n is odd, adding equations (9) and (10) gives

2E(G) = (−1)^{n+1} ∑_{i}(−1)^{i}[F_{i}],

and hence the claim.

Let J be a non-cyclic finite subgroup of the rotation group SO(3). Recall that the natural action of J on the 2-sphere has three non-free orbits. Let H_{1}, H_{2} and H_{3} be stabilizers of points in these three orbits.

Corollary 24. For J, H_{i} as above, let G be the double of J along the H_{1}, H_{2}, H_{3}, and let R be a ring in which |J| is a unit. Then G is a virtually-free group of type FPP over R and the Wall obstruction Ẽ(G) in K̃_{0}(RG) has order at most two.

Proof. We’ll build a contractible 3-manifold with a G-action satisfying the conditions of Theorem 23. View the 3-sphere as the 1-point compactification of ℝ^{3}. The usual SO(3) action on ℝ^{3} extends to an action on the 3-sphere with fixed points at 0 and ∞. Let B_{1} be the closed unit ball in ℝ^{3} centred at 0, and let B_{2} be the complement in the 3-sphere of the open ball of radius two, so that B_{2} is a ball centred at ∞. Pick points x_{i} for i = 1, 2, 3 on the surface of B_{1} in distinct Jorbits, such that the stabilizer of x_{i} is H_{i}. Finally, for each i, let C_{i} be a cylinder joining B_{1} and B_{2} with x_{i} on its axis. Pick a J-equivariant triangulation for the union of B_{1}, B_{2} and the images under J of the C_{i}. (The radius of C_{i} should be sufficiently small that these images are disjoint.)

This gives J-spaces B_{j} for j = 1, 2 and for each i an H_{i}-space C_{i}, together with H_{i}-equivariant attaching maps from one end of C_{i} to the boundary of B_{j} for each i and j. Moreover, the singular set of B_{j} (which is the central point together with a collection of intervals meeting at the centre) is a J-equivariant retract of B_{j}, the singular set of C_{i} (which is its central line) is an H_{i}-equivariant retract of C_{i}, and these retractions can be chosen to be compatible with the gluing of C_{i} to B_{j}. Now view B_{j} as a J_{j}-space for j = 1, 2.

Each of the five spaces constructed so far can be ‘induced up to G’, to give G-spaces G ×_{J_{j}} B_{j} and G ×_{H_{i}} C_{i}, and the H_{i}-equivariant attaching maps give rise to G-equivariant attaching maps between these five induced G-spaces. The resulting G-space X has a G-tree as an equivariant retract (the tree corresponding to the given expression for G as a graph of groups), and so X is contractible. Moreover, X is a 3manifold with boundary, and G acts freely on δX. Hence Theorem 23 applies to X, and the claim follows.

The next proposition completes the proof of Theorem 3.

Proposition 25. Let J be a finite subgroup of SO(3), and construct G as in Corollary 24. If J is D_{n} for n even, or A_{4}, S_{4} or A_{5}, then Ẽ(G) ∈ K̃_{0}(ℂG) has order two. If J = D_{n} for n odd, then Ẽ(G) ∈ K̃_{0}(ℂG) is zero, but Ẽ(G) ∈ K̃_{0}(ℚG) has order two.

Proof. By Corollary 24, the order of Ẽ(G) is at most two. Since K̃_{0}(kJ) is free abelian, it follows from Proposition 8 that Ẽ(G) = 0 ∈ K̃_{0}(kG) if and only if the trivial kJ-module is a sum of induced modules from the H_{i}. If J is the symmetry group of a platonic solid, or D_{n} for n even, then each of the H_{i} has even index in J. Thus in this case any induced module has even k-dimension, so the trivial module is not a sum of induced modules.

If J = D_{n} for n odd, the trivial ℂC_{2}-module induces up to J as the sum of the trivial ℂJ-module direct sum one copy of each irreducible ℂJ-module of dimension two. Since each irreducible 2-dimensional ℂJmodule is induced up from C_{n}, it follows that the trivial ℂJ-module is a sum of induced modules as required.

It remains to check that when J = D_{n} for some odd n, the trivial ℚJ-module is not a sum of induced modules (from proper subgroups). In the case when n is an odd prime p, there are only three irreducible ℚJ-modules, and it can be checked that induced modules (from any proper subgroup) consist of an even number of ℚJ-irreducibles. This proves the claim when n = p. The claim for general odd n follows, since if p divides n, there is a surjective homomorphism from D_{n} to D_{p}, which takes the trivial ℚD_{n}-module to the trivial ℚD_{p}-module.

9. Comparing FL and FH

M. Bestvina and N. Brady define G to be of type FH over R if there is an R-acyclic, G-free G-CW-complex Y with finitely many orbits of cells [5]. Using the simplicial approximation theorem it may be shown that any G-CW-complex Y is G-homotopy equivalent to a simplicial complex X on which G acts simplicially. Moreover, if Y has finitely many orbits of cells, X has finitely many orbits of simplices. Thus there is no loss of generality if one assumes that the space Y is a simplicial complex.

If G is FH over R, the chain complex for the space Y is an RG-free resolution for the trivial module R, and hence G is FL over R. The converse does not hold for all choices of G and R however, as we shall see below.

If K is an extension field of a field k, then for any space X, and any i, the homology group H_{i}(X; K) is naturally isomorphic to H_{i}(X; k)⊗K. In particular, X is k-acyclic if and only if X is K-acyclic. Thus one obtains the following proposition.

Proposition 26. Let K be an extension field of k. A group G is FH over K if and only if G is FH over k.

Corollary 27. There are virtually-free groups and PD^{3}-groups over the complex numbers ℂ that are FL over ℂ but not FH over ℂ.

Proof. By Proposition 26, any group that is FL over ℂ but is not FL over ℚ cannot be FH over ℂ. Virtually-free groups having these properties are given in Proposition 25 and PD^{3}-groups having these properties in Theorems 14 and 19.

To demonstrate that this corollary is interesting, we ought to show that many groups of type FL are also of type FH. This is the content of Theorem 29.

A CW-complex is said to be of finite type if it has finitely many cells in each dimension. (Note that a simplicial complex with finitely many vertices must be finite, so for this finiteness condition one really must consider CW-complexes instead.) An Eilenberg-Mac Lane space for a group G, denoted K(G, 1), is a connected CW-complex whose fundamental group is G and whose universal cover is contractible. The proposition below shows that every group considered in this paper has a finite-type Eilenberg-Mac Lane space.

Proposition 28. If G is virtually torsion-free and acts with finite stabilizers and finitely many orbits on a contractible simplicial complex X, then G has a finite-type Eilenberg-Mac Lane space.

Proof. Let H be a torsion-free finite-index normal subgroup of G, and let Y be the universal cover of a finite-type Eilenberg-Mac Lane space for G/H. (The ‘bar construction’ shows that any finite group has a finite type Eilenberg-Mac Lane space [9].) Replacing X by its barycentric subdivision if necessary, it may be assumed that the stabilizer of every simplex of X fixes that simplex point-wise. Now X may be viewed as a G-CW-complex whose cells just happen to be simplexshaped. Also X×Y is a contractible G-CW-complex with finitely many orbits of cells of each dimension. Since the stabilizer of any cell of Y is H, and H acts freely on X, it follows that G acts freely on X × Y, and the quotient (X × Y)/G is a finite-type K(G, 1).

Theorem 29. Let R be a subring of the rationals, and let G be a group with an Eilenberg-Mac Lane space of finite type. Then G is FL over R if and only if G is FH over R.

Proof. Without any conditions on G and R, FH ⇒ FL, but we must prove the converse. Let Y be the universal cover of a finite-type Eilenberg-Mac Lane space for G. Let n be the maximum of 3 and the cohomological dimension of G over R, and let X′ be the (n − 1)- skeleton of Y. The R-chain complex C′_{∗} for X′ is the start of a finite RG-free resolution for R. Hence

H′ = H_{n−1}(X′; R) = ker(C′_{n−1} → C′_{n−2})

is a stably-free RG-module, i.e., there exists m such that H′ ⊕ RG^{⊕m} is free. Build X′′ by attaching m free G-orbits of (n − 1)-spheres to X′. (Equivalently one could attach a bouquet of m copies of the (n − 1)- sphere to Y/G, and define X′′ to be the (n−1)-skeleton of the universal cover of this space.) Now X′′ is an (n − 1)-dimensional G-free G-CW-complex which is (n − 2)-connected, and has the property that H′′ = H_{n−1}(X′′; R) is a free RG-module, with RG-basis h_{1}, …, h_{l} say.

H′′ is a submodule of the (n − 1)-chains on X′′, so each h_{i} may be represented as a finite sum h_{i} = ∑_{j} r_{ij}c_{j} for r_{ij} ∈ R and c_{j} an (n − 1)- cell of X′′. Since R is a subring of ℚ, there is an integer s_{i} which is a unit in R and such that for each j, s_{i}r_{ij} ∈ ℤ. (Take s_{i} to be the l.c.m. of the denominators of the r_{ij} when expressed as quotients of coprime integers.) The elements s_{i}h_{i} form an RG-basis for H′′, and each s_{i}h_{i} lies in the integral (n−1)-chains on X′′. Since X′′ is (n−2)-connected, there is a map f_{i} from the (n − 1)-sphere to X′′ sending a generator for the top homology group of the sphere to s_{i}h_{i}. Build X by attaching

THE EULER CLASS OF A POINCARÉ DUALITY GROUP 25

l free G-orbits of n-cells to X′′ using the f_{i} as attaching maps for a set of orbit representatives. This X is the required R-acyclic G-free G-CW-complex.

10. Detecting K_{0}

In this section we give a proof that certain elements of K_{0} have order two, that does not depend on [2, 11, 24, 25]. This gives a self-contained and comparatively elementary proof of parts (a) and (b) of Theorem 3, together with some cases of Theorem 2. A different account of this argument has appeared in [15], so we omit some of the calculations.

Let f be the embedding of ℂ in M_{2}(ℝ) defined by

f : re^{iθ} ↦ ( \begin{matrix}r cos θ & −r sin θ \\ r sin θ & r cos θ \end{matrix} ) .

Let S_{1} and S_{2} be copies of M_{2}(ℝ), and let R be the coproduct of S_{1} and S_{2}, amalgamating the images of ℂ. As a ℂ-bimodule, M_{2}(ℝ) is freely generated by

I = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) and A = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

It follows that R has ℂ-basis the alternating words in A and A′, the copies of A lying in S_{1} and S_{2} respectively. Thus R may be identified with the skew group ring ℂ ∗ D, where D is an infinite dihedral group generated by A and A′, and AλA = A′λA′ = λ^{∗}, the complex conjugate of λ, for each λ ∈ ℂ.

The map

1 ⊗ f : ℂ ⊗ ℂ → ℂ ⊗ M_{2}(ℝ),

where all tensor products are over ℝ, is the inclusion of a subring isomorphic to ℂ×ℂ in M_{2}(ℂ). Hence ℂ⊗R may be identified with the coproduct of two copies of M_{2}(ℂ), amalgamating a subring isomorphic to ℂ×ℂ. In terms of the description of R as a skew group algebra, ℂ⊗R is isomorphic to (ℂ × ℂ) ∗ D, where A(λ, μ)A = A′(λ, μ)A′ = (μ, λ) for any (λ, μ) ∈ ℂ×ℂ. The ℝ-algebra R is one of the examples mentioned at the start of Section 6.

Now let e be a rank one idempotent in M_{2}(ℝ), and let e, e′ denote copies of e in S_{1} and S_{2} respectively. Let [e] and [e′] denote the corresponding elements of K_{0}(R). In K_{0}(M_{2}(ℝ)), 2[e] is equal to the free module of rank one. It follows that 2([e] − [e′]) = [R] − [R] = 0 in K_{0}(R).

On the other hand, the image of 1 ⊗ f : ℂ ⊗ ℂ → M_{2}(ℂ) contains rank one idempotents. It follows that [e] − [e′] = 0 in K_{0}(ℂ ⊗ R). To show that [e] − [e′] is an element of order two in K_{0}(R) that is mapped to zero in K_{0}(ℂ ⊗ R) it suffices to show

Theorem 30. The elements [e] and [e′] are not equal in K_{0}(R).

Proof. Let T be the ring of real polynomial functions on the circle x^{2} + y^{2} = 1. Parametrizing the circle by t ∈ ℝ/ℤ, T is generated as an ℝ-algebra by c and s defined as:

c(t) = cos(2πt), s(t) = sin(2πt).

The ring M_{2}(T) may be viewed as a ring of polynomial functions from the circle to M_{2}(ℝ). Define homomorphisms

f_{i} : S_{i} → M_{2}(T)

by

f_{1}(M) = (t ↦ M) , f_{2}(M) = (t ↦ f(e^{πit})Mf(e^{−πit})) ,

for any M ∈ S_{i}, where f : ℂ → M_{2}(ℝ) is as above. Since the image of f is commutative, f_{1} and f_{2} agree on the images of ℂ in S_{1} and S_{2}, and hence define a homomorphism f̂ : R → M_{2}(T). In fact, f̂ is an isomorphism from R to M_{2}(T).

The homomorphisms f_{1} and f_{2} satisfy

f_{1} : ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ↦ ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ,

f_{2} : ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ↦ ( \begin{matrix}(1 + c)/2 & s/2 \\ s/2 & (1 − c)/2 \end{matrix} ) .

Hence the images of [e] and [e′] in K_{0}(T) (under the composite of f and the Morita equivalence map) are the classes represented by the free module of rank one and the ideal (c + 1, s) respectively. Since (c + 1, s) is a non-principal maximal ideal in the Dedekind domain T, it is not stably isomorphic to the free module (see 1.4.23 of [23]).

This argument is similar to that used in [15], except that there a topological argument (via the ring of continuous functions on the circle) was used to detect [e] − [e′].

Let G be the double of the dihedral group D_{p} of order 2p along its subgroup C_{p}, for p an odd prime. Recall that in Proposition 16 it was shown that

K_{0}(ℝG) ≅ ℤ^{⊕(p+5)/2} ⊕ ℤ_{2}^{⊕(p−1)/2}.

The rank of K_{0}(ℝG) is equal to the number of conjugacy classes of elements of finite order in G, and so Hattori-Stallings traces detect K_{0}(ℝG) modulo its torsion subgroup. The torsion subgroup of K_{0}(ℝG)

THE EULER CLASS OF A POINCARÉ DUALITY GROUP 27

can be detected by homomorphisms from ℝG to the ring R of Theorem 30, as follows.

Let ρ be an irreducible 2-dimensional real representation of the dihedral group D_{p}. Thus ρ gives rise to a homomorphism ρ : ℝD_{p} → M_{2}(ℝ). It may be assumed that ρ is orthogonal, in which case ρ(C_{p}) is contained in f(ℂ) ⊆ M_{2}(ℝ). The corresponding ℝ-algebra homomorphisms

ρ_{1} : ℝD_{p} → S_{1} ⊆ R ρ_{2} : ℝD_{p} → S_{2} ⊆ R

coincide on ℝC_{p}, and hence induce a homomorphism

ρ̂: ℝG → R.

There are (p − 1)/2 different choices of ρ. As an ℝ-algebra, ℝD_{p} is isomorphic to ℝ × ℝ × M_{2}(ℝ)^{⊕(p−1)/2}, and the different choices of ρ correspond to projections on to the M_{2}(ℝ) factors. This gives rise to (p − 1)/2 distinct homomorphisms from ℝG to R, and hence (p − 1)/2 homomorphisms from K_{0}(ℝG) to K_{0}(R).

There is a ℤ_{2}-basis for the torsion subgroup of K_{0}(ℝG) consisting of the elements [e_{i}] − [e′_{i}], 1 ≤ i ≤ (p − 1)/2, where e_{i} is a primitive idempotent in ℝD_{p} lying in the ith copy of M_{2}(ℝ), and e_{i}, e′_{i} denote corresponding elements in the two copies of ℝD_{p} inside ℝG. Each of these elements is sent to the element of order two in K_{0}(R) by exactly one of the (p − 1)/2 homomorphisms from ℝG to R, and is sent to zero by the others. Hence the whole torsion subgroup of K_{0}(ℝG) is detected by these homomorphisms.

In the cases when either k is a subfield of ℝ, or k contains a primitive pth root of unity, this gives an alternative method to detect all of the elements of K_{0}(kG) given in Proposition 16. This suffices to prove cases (a) and (b) of Theorem 3.

11. Stably-free modules

In this section, we exhibit non-free stably-free finitely generated kGmodules for a certain virtually-free group G and any field k of characteristic zero. Note that H. Bass has shown that when G is free, every projective kG-module is free [1]. (This can also be deduced from the theorem of Bergman-Dicks quoted below as Theorem 31.) See [13, 3, 16] for examples of non-free stably-free modules for group algebras of certain torsion-free groups.

For any ring R, let P(R) denote the abelian monoid of isomorphism types of finitely generated projective R-modules. A projective R-module P is stably-free of rank r but not free if and only if [P] = r[R] in K_{0}(R), but [P] ≠ r[R] in P(R). As remarked earlier, the Bergman-Dicks approach to Theorem 7, which has the drawback that it requires stronger hypotheses than Waldhausen’s, does have the advantage that it also gives the following.

Theorem 31. Let G be the fundamental group of a finite graph of finite groups, and let C be a category with objects the edge and vertex groups in the graph for G, with (non-identity) morphisms the inclusions of each edge group in its initial and terminal vertex groups. Let k be a field in which the order of each of the finite groups in C is a unit. Then the following natural map is an isomorphism:

colim_{H∈C} P(kH) → P(kG).

The corresponding statement for graphs of groups with larger edge groups is not valid. Lewin exhibits non-free stably-free ℚG-modules for a class of groups which includes the fundamental groups of the trefoil knot and of the Klein bottle [16]. Dunwoody’s non-free stably-free ℤGmodules for G the trefoil group [13] also give rise to non-free stably-free ℚG-modules for this G. On the other hand, the trefoil and Klein bottle groups may be expressed as fundamental groups of graphs of groups with infinite cyclic edge and vertex groups, and every projective module for the rational group algebra of an infinite cyclic group is free.

Corollary 32. Let G be the double of the alternating group J = A_{5} along subgroups H_{1} and H_{2} of order two and three. For any field k of characteristic zero there is a finitely generated projective kG-module P such that P ⊕ kG is free, but for any n > 0, P^{⊕n} is not free.

Proof. For any finite group L, the monoid P(kL) is isomorphic to a direct sum of copies of ℕ indexed by the simple kL-modules, and embeds in K_{0}(kL). For each choice of k, we shall exhibit finitely generated (projective) kJ-modules M_{1} and M_{2} with the following properties.

1. For some r > 0, [M_{1}] + [M_{2}] = r[kJ] in P(kJ);

2. For each n > 0 and each 1 ≤ i, j ≤ 2, if N ≤ (M_{i})^{⊕n} and N is isomorphic to a module induced up from H_{j}, then N = 0;

3. In K_{0}, for each 1 ≤ i ≤ 2, [M_{i}] is in the image of the induction

map K_{0}(kH_{1}) ⊕ K_{0}(kH_{2}) → K_{0}(kJ). Given such M_{i}, let J_{1} and J_{2} be the two copies of J used to build G, and define P = Ind_{J_{1}}^{G}(M_{1}) ⊕ Ind_{J_{2}}^{G}(M_{2}). Then by (1) and (3), it follows that in K_{0}(kG),

[P] = Ind_{J_{1}}^{G}([M_{1}] + [M_{2}]) = Ind_{J_{1}}^{G}(r[kJ_{1}]) = r[kG].

On the other hand, the description of P(kG) given by Theorem 31 and property (2) imply that [P] ≠ r[kG] in P(kG).

It remains to exhibit M_{1}, M_{2} satisfying (1)–(3). Firstly, suppose that k is a splitting field for J and for the H_{j}. (This occurs if and only if k contains ω, a primitive cube root of 1, and \sqrt{5}.) In this case, there are five simple kJ-modules, which we shall denote by χ_{1}, …, χ_{5}, of k-dimensions 1, 3, 3, 4, and 5 respectively. The modules

M_{1} = χ_{1} + χ_{4} + 5χ_{5}, M_{2} = 3χ_{2} + 3χ_{3} + 3χ_{4}

have properties (1)–(3), and give rise to a non-free stably-free kGmodule of rank one. The same elements will suffice if k contains ω but does not contain \sqrt{5}, except that in this case there are only four simple kJ-modules, and χ_{2} + χ_{3} should be understood to denote a simple kJmodule of dimension six. When k does not contain ω, the M_{i} as above do not satisfy (3). However, the modules M′_{i} = M_{i} ⊕ M_{i} satisfy (1)– (3) and may be used to construct a kG-module that is not free and is stably-free of rank two.

Of course the method used in proving Corollary 32 could be applied to the fundamental groups of many other graphs of finite groups, but we preferred to concentrate on a specific example.

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Faculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ, United Kingdom

  E-mail address: I.J.Leary@maths.soton.ac.uk