TOPOLOGICAL EQUIVALENCE OF LINEAR REPRESENTATIONS FOR CYCLIC GROUPS: II

IAN HAMBLETON AND ERIK K. PEDERSEN

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

1. Introduction

Let G be a finite group and V, V′ finite dimensional real orthogonal representations of G. Then V is said to be topologically equivalent to V′ (denoted V ∼_{t} V′) if there exists a homeomorphism h: V → V′ which is G-equivariant. If V, V′ are topologically equivalent, but not linearly isomorphic, then such a homeomorphism is called a non-linear similarity. These notions were introduced and studied by de Rham [23], [24], and developed extensively in [1], [2], [14], [15], and [5]. In the two parts of this paper, referred to as [I] and [II], we complete de Rham’s program by showing that Reidemeister torsion invariants and number theory determine non-linear similarity for finite cyclic groups.

A G-representation is called free if each element 1 ≠ g ∈ G fixes only the zero vector. Every representation of a finite cyclic group has a unique maximal free subrepresentation.

Theorem. Let G be a finite cyclic group and V_{1}, V_{2} be free G–representations. For any G–representation W, the existence of a non-linear similarity V_{1} ⊕ W ∼_{t} V_{2} ⊕ W is entirely determined by explicit congruences in the weights of the free summands V_{1}, V_{2}, and the ratio Δ(V_{1})/Δ(V_{2}) of their Reidemeister torsions, up to an algebraically described indeterminacy.

The notation and the indeterminacy are given in Section 2 and a detailed statement of results in Theorems A–E. This part of the paper contains the foundational results and calculations in bounded algebraic K- and L-theory needed to prove the main results on non-linear similarity. The study of non-linear similarities V_{1} ⊕ W ∼_{t} V_{2} ⊕ W increases in difficulty with the number of isotropy types in W. We introduce a new method using excision in bounded surgery theory, based on the orbit type filtration, to organize and deal with these difficulties. We expect that this technique will be useful for other applications. Our most general results about non-linear similarity for arbitrary cyclic groups are Theorem C and its extensions (see Sections 9 and 10).

In Sections 3 and 13 we study the group R_{Top}(G) of G–representations modulo stable topological equivalences (see [2] where R_{Top}(G) ⊗ Q is computed). As an application of our general results, we determine the structure of the torsion in R_{Top}(G), for G any cyclic group (see Theorem 13.1), and in Theorem D we give the calculation of R_{Top}(G) for G = C(4q), for q odd, correcting [5, Thm. 2]. One interesting feature is that Corollary 2.4 and Theorem D indicate a connection between the orders of the ideal class groups for cyclotomic fields and topological equivalence of linear representations.

Date: Feb. 19. 2004.

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

Contents

1. Introduction 1 2. Statement of Results 2 3. A splitting of R_{Top}(G) 7 4. A rational computation 9 5. Excision in bounded surgery theory 12 6. Calculations in bounded K-theory 17 7. The double coboundary 22 8. Calculations in bounded L–theory 25 9. The proof of Theorem C 30 10. Iterated R_{−} transfers 32 11. The proof of Corollary 2.4 36 12. The normal invariant 37 13. The proof of Theorem D 40 References 42

2. Statement of Results

For the reader’s convenience, we recall some notation from Part I, and then give the main results of both parts. Theorems A and B are proved in Part I and Theorems C and D are proved in Part II. The proof of Theorem E is divided between the two parts.

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

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

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

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

Let V_{2} = t^{b_{1}} + ⋯ + t^{b_{k}} be another free representation, such that S(V_{1}) and S(V_{2}) are Ghomotopy equivalent. This just means that the products of the weights ∏ a_{i} ≡ ∏ b_{i} mod 4q.

Then the Whitehead torsion of any G-homotopy equivalence is determined by the element

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

since Wh(ZG) → Wh(QG) is monic [17, p.14].

Let W be a finite-dimensional G-representation. A necessary condition for a non-linear similarity V_{1}⊕W ∼_{t} V_{2}⊕W is the existence of a G-homotopy equivalence f: S(V_{2}) → S(V_{1}) such that f ∗ id: S(V_{2} ⊕ U) → S(V_{1} ⊕ U) is freely G-normally cobordant to the identity map on S(V_{1} ⊕ U), for all free G–representations U (see [I], Section 3). If V_{1} and V_{2} satisfy this condition, we say that S(V_{1}) and S(V_{2}) are s–normally cobordant. This condition for non-linear similarity can be decided by explicit congruences in the weights of V_{1} and V_{2} (see [30, Thm. 1.2]).

This quantity, Δ(V_{1})/Δ(V_{2}) is the basic invariant determining non-linear similarity. It represents a unit in the group ring ZG, explicitly described for G = C(2^{r}) by Cappell and Shaneson in [3, §1] using a pull-back square of rings. To state concrete results we need to evaluate this invariant modulo suitable indeterminacy.

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

{Δ(V_{1})/Δ(V_{2})} ∈ H^{0}(Wh(ZG))

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

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

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

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

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

This calculation takes place over the ring Λ_{2q} = Z[t]/(1 + t^{2} + ⋯ + t^{4q−2}), but the result holds over ZG via the involution–invariant pull-back square

ZG →  Λ_{2q}
↓  ↓
Z[Z/2] → Z/2q[Z/2]

Consider the exact sequence of modules with involution:

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

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

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

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

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

Our first result deals with isotropy groups of index 2, as is the case for all the non–linear similarities constructed in [1].

Theorem A. Let V_{1} = t^{a_{1}} + ⋯ + t^{a_{k}} and V_{2} = t^{b_{1}} + ⋯ + t^{b_{k}} be free G–representations, with a_{i} ≡ b_{i} ≡ 1 mod 4. There exists a topological similarity V_{1} ⊕ R_{−} ∼_{t} V_{2} ⊕ R_{−} if and only if

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

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

Remark 2.2. The proof of this result is in Part I, but note that Condition (iii) simplifies for G a cyclic 2-group since H^{0}(k) = 0 in that case (see [I], Lemma 9.1). Theorem A should be compared with [1, Cor.1], where more explicit conditions are given for “firsttime” similarities of this kind under the assumption that q is odd, or a 2-power, or 4q is a “tempered” number. See also Theorem 9.2 for a more general result concerning similarities without R_{+} summands. The case dim V_{1} = dim V_{2} = 4 gives a reduction to number theory for the existence of 5-dimensional similarities (see [I], Remark 7.2).

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

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

and consider the exact sequence

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

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

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

There is a natural map H^{1}(Wh(ZG^{−})/ Wh(ZH)) → H^{1}(Wh(ZH → ZG^{−})). We will use the same notation

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

to denote the image of our Reidemeister torsion invariant. The non-linear similarities handled by the next result have isotropy of index ≤ 2.

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

(i) ∏ a_{i} ≡ ∏ b_{i} mod 4q, (ii) Res_{H} V_{1} ≅ Res_{H} V_{2}, and (iii) the element {Δ(V_{1})/Δ(V_{2})} is in the image of the double coboundary

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

This result can be applied to 6-dimensional similarities.

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

Remark 2.5. For example, the class number condition is satisfied for q ≤ 11, but not for q = 29. The proof of the Corollary 2.4 is given in Section 11 assuming Theorem B, which is proved in Part I. This result corrects [5, Thm.1(i)], and shows that the computations of R_{Top}(G) given in [5, Thm. 2] are incorrect.

Our final example of the computation of bounded transfers is suitable for determining stable non–linear similarities inductively, with only a minor assumption on the isotropy subgroups. To state the algebraic conditions, we must again generalize the indeterminacy for the Reidemeister torsion invariant to include bounded K–groups (see Section 5). In this setting K̃_{0}(ZH →ZG) = K̃_{0}(C_{R_{−},G}(Z)) and Wh(ZH →ZG) = Wh(C_{R_{−},G}(Z)). We consider the analogous double coboundary

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

and note that there is a map Wh(C_{R_{−},G}(Z)) → Wh(C_{W ×R_{−},G}(Z)) induced by the inclusion on the control spaces. We will again use the same notation

{Δ(V_{1})/Δ(V_{2})} ∈ H^{1}(Wh(C_{W ×R_{−},G}(Z)))

for the image of the Reidemeister torsion invariant in this new domain.

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

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

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

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

Remark 2.6. The existence of a similarity V_{1} ⊕ W ∼_{t} V_{2} ⊕ W implies that S(V_{1}) and S(V_{2}) are s-normally cobordant. In particular, S(V_{1}) must be freely G-normally cobordant to S(V_{2}) and this (unstable) normal invariant condition is enough to give us a surgery problem. Crossing with W defines the bounded transfer map

trf_{W} : L_{2k}^{h}(ZG) → L_{2k+dim W}^{h} (C_{W,G}(Z))

introduced in [10]. The vanishing of the surgery obstruction is equivalent to the existence of a similarity (see [I], Theorem 3.5). The computation of the bounded transfer in L-theory leads to condition (iii), and an expression of the obstruction purely in terms of bounded K-theory. To carry out this computation we may need to stabilize in the free part, and this uses the s-normal cobordism condition.

Remark 2.7. Note that W_{max} = 0 in condition (iii) if W has no isotropy subgroups of 2-power index. Theorem C suffices to handle stable topological similarities, but leaves out cases where W has an odd number of R_{−} summands (handled in Theorem 9.2 and the results of Section 10). Simpler conditions can be given when G = C(2^{r}) (see [I], Section 9).

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

(2.8)  0 → Wh(C_{R_{−},G}(Z)) → Wh(C_{W_{max}×R_{−},G}(Z)) → K_{1}(C_{W_{max}×R_{−},G}^{>R_{−}}(Z)) → 0

derived in Corollary 6.9. We have K_{1}(C_{W_{max}×R_{−},G}^{>R_{−}}(Z)) = K_{−1}(ZK), where K is the isotropy group of W_{max}, and Wh(C_{R_{−},G}(Z)) = Wh(ZH → ZG). The indeterminacy in Theorem C is then generated by the double coboundary

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

used in Theorem B and the coboundary

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

from the Tate cohomology sequence of (2.8).

Finally, we apply these results to R_{Top}(G). Since its rank is known (see [2] or Section 4), it remains to determine its torsion subgroup. In Section 3, we will define a filtration

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

on the real representation ring R(G), inducing a filtration on R_{Top}(G) = R(G)/R_{t}(G). Here the subgroup

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

is generated by stable topological similarity. Note that R(G) has the following nice basis: {t^{i}, δ, ϵ | 1 ≦ i ≦ 2q − 1}, where δ = [R_{−}] and ϵ = [R_{+}] (although we do not have i ≡ 1 mod 4 for all the weights).

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

R(G) = ⨁_{K⊆G} R^{free}(G/K)

and this direct sum splitting can be intersected with R_{t}(G) to define Rt^{free}(G). We let RTop^{free}(G) = R^{free}(G)/Rt^{free}(G). Since inflation and fixed sets preserve topological similarities, we obtain an induced splitting

RTop^{free}(G) = ⨁_{K⊆G} RTop^{free}(G/K) .

By induction on the order of G, we see that it suffices to study the summand RTop^{free}(G).

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

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

In Section 4 we prove that R̃Top^{free}(G) is precisely the torsion subgroup of RTop^{free}(G). Here is a specific computation (correcting [5, Thm. 2]).

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

For any cyclic group G, we use normal cobordism and homotopy equivalence to define a filtration

Rt^{free}(G) ⊆ Rn^{free}(G) ⊆ Rh^{free}(G) ⊆ R^{free}(G)

leading by direct sum to the filtration of R(G) mentioned above. Both R̃^{free}(G)/R̃h^{free}(G) and R̃h^{free}(G)/R̃n^{free}(G) are torsion groups which can be explicitly determined by congruences in the weights (see Section 12 and [30, Thm.1.2]). The subquotient R̃n^{free}(G)/Rt^{free}(G) always has exponent two (see Section 13).

We conclude this list of sample results with a calculation of R_{Top}(G) for cyclic 2-groups (see [I] for the proof).

Theorem E. Let G = C(2^{r}), with r ≧ 4. Then

R̃Top^{free}(G) = ⟨α_{1}, α_{2}, …, α_{r−2}, β_{1}, β_{2}, …, β_{r−3}⟩

subject to the relations 2^{s}α_{s} = 0 for 1 ≦ s ≦ r − 2, and 2^{s−1}(α_{s} + β_{s}) = 0 for 2 ≦ s ≦ r − 3, together with 2(α_{1} + β_{1}) = 0.

The generators for r ≧ 4 are given by the elements

α_{s} = t − t^{5^{2^{r−s−2}}} and β_{s} = t^{5} − t^{5^{2^{r−s−2}+1}}.

We remark that R̃Top^{free}(C(8)) = Z/4 generated by t − t^{5}. In [I], Theorem 11.6 we use this information to give a complete topological classification of linear representations for cyclic 2-groups.

Acknowledgement. The authors would like to express their appreciation to the referee for many constructive comments and suggestions. In particular, the referee pointed out an improvement to the statement of Theorem E. If we let ψ_{i} = α_{i+1}+β_{i+1} for 1 ≤ i ≤ r−4, then R̃Top^{free}(C(2^{r})) is a direct sum of cyclic groups generated by α_{1}, …, α_{r−2}, β_{1} and ψ_{1}, …, ψ_{r−4} where the order of the cyclic group generated by a basis element with subscript i is 2^{i}. This basis displays the group structure explicitly.

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

In this section we point out an elementary splitting of R_{Top}(G), and some useful filtrations. For G any finite group, we denote by R(G) the real representation ring of G. Elements in R(G) can be given as formal differences (V_{1} − V_{2}) of G–representations, and (V_{1} − V_{2}) ∼ 0 if and only if there exists a representation W such that V_{1} ⊕ W ≅ V_{2} ⊕ W.

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

inf_{K}: R(G/K) → R(G)

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

Fix_{K}: R(G) → R(G/K)

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

Definition 3.1. A G–representation V is free if V^{K} = {0} for all non–trivial normal subgroups 1 ≠ K ◁ G.

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

(3.2)  R^{free}(G) = ⋂{ker Fix_{K} | 1 ≠ K ◁ G}

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

Proposition 3.3. There is a direct sum splitting

R(G) = ⨁_{K◁ G} R^{free}(G/K)

indexed by the normal subgroups K in G.

Proof. Let V (K) denote the G–invariant subspace given by the sum of all the irreducible sub–representations of V with kernel exactly K. This is a free G/K representation. The decomposition above is given by mapping (V_{1} − V_{2}) to the elements (V_{1}(K) − V_{2}(K)). □

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

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

and the quotient group is R_{Top}(G) by definition. We define Rt^{free}(G) = R^{free}(G) ∩ R_{t}(G). Since R_{t}(G) is preserved by inflation and taking fixed sets, we obtain

Corollary 3.5. There is a direct sum decomposition

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

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

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

(3.6)  Rh^{free}(G) = {(V_{1} − V_{2}) | S(V_{1}) ≃_{G} S(V_{2}) for V_{1} and V_{2} free}

where ≃_{G} denotes G–homotopy equivalence. This is a subgroup of R^{free}(G), in fact a sub–Mackey functor since it has induction and restriction for subgroups of G. We define

(3.7)  R_{h}(G) = ⨁_{K◁ G} Rh^{free}(G/K)

If there exists a G–homotopy equivalence f: S(V_{1}) → S(V_{2}) such that

S(V_{1} ⊕ U) = S(V_{1})∗S(U) ⟶^{f∗1} S(V_{2})∗S(U) = S(V_{2} ⊕ U)

is freely G–normally cobordant to the identity for all free G–representations U, then we say that S(V_{1}) and S(V_{2}) are s-normally cobordant, and we write S(V_{1}) ≏_{G} S(V_{2}). Define

(3.8) Rn^{free}(G) = {α ∈ Rh^{free}(G) | ∃V_{1}, V_{2} with α = (V_{1} − V_{2}) and S(V_{1}) ≏_{G} S(V_{2})}

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

(3.9)  R_{n}(G) = ⨁_{K◁ G} Rn^{free}(G/K)

Since Rt^{free}(G) ⊆ Rn^{free}(G), we have defined a filtration

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

of R(G), natural with respect to restriction of representations. All the terms except possibly R_{n}(G) are also natural with respect to induction of representations.

Remark 3.11. It follows from the proof of [30, 3.1] that S(V_{1}⊕U_{0}) is s-normally cobordant to S(V_{2}⊕U_{0}), for some free G-representation U_{0}, if and only if S(V_{1}) is s-normally cobordant to S(V_{2}). It follows that we could have used the latter condition to define Rn^{free}(G).

4. A rational computation

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

First we consider cyclic groups. Let G = C(2^{r}q) be a cyclic group, where q ≧ 1 is odd. The Odd Order Theorem [14], [15] gives R(G) = R_{Top}(G) if r ≦ 1, and we recall the definition

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

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

Theorem 4.1. For G cyclic, the kernel and cokernel of

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

are 2–primary torsion groups.

Corollary 4.2. Let G = C(2^{r}q), q odd, be a finite cyclic group.

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

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

(iii) We have the formula

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

where φ(d) is the Euler function.

Proof. The first part follows directly from Theorem 4.1. For parts (ii) and (iii) count the free representations of G_{odd}. □

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

Lemma 4.3. For G cyclic, the composition of the natural map R_{F} (G) → R(G) and Res: R(G) → R(G_{odd}) induces a p–local isomorphism RF^{free}(G) → R^{free}(G_{odd}), for any odd prime p.

Proof. According to a result of Brauer [25, Thm.24] R_{F} (G_{odd}) = R(G_{odd}), and any representation of G can be realized over the field Q(ζ_{|G|}). In addition, the restriction map

Res: R_{F} (G) → R_{F} (G_{odd})

is a p–local surjection (since Res_{G_{odd}} ∘ Ind_{G_{odd}} is just multiplication by 2^{r}). But the rank of R_{F} (G) given in [25, 12.4] equals the rank of R(G_{odd}), so we are done. □

Corollary 4.4. For G cyclic and any odd prime p, the natural map R_{F} (G) → R_{Top}(G) induces a p–local isomorphism.

Proof. This follows from the Lemma 4.3 and Theorem 4.1. □

In [2], Cappell and Shaneson obtained the following result by a different argument. It computes the rank of R_{Top}(G) ⊗ Q for any finite group G.

Theorem 4.5. Let F be the composite of all subfields of C of the form Q(ζ + ζ^{−1}) where ζ ∈ C is an odd root of unity. Then for any finite group G, the natural map R_{F} (G) → R(G) induces an isomorphism R_{F} (G) ⊗ Z_{(p)} ≅ R_{Top}(G) ⊗ Z_{(p)} for any odd prime p.

Proof. The result holds for cyclic groups G by Corollary 4.4, and we apply induction theory to handle general finite groups.

First we observe that the Mackey functors R(G) ⊗ Z_{(p)} and R_{F} (G) ⊗ Z_{(p)} are generated by induction from p–elementary subgroups (see [25, Thm.27] and note that Γ_{F} = {±1}). Therefore, by [17, 11.2] they are p–elementary computable in the sense of Dress induction theory [9]. We may therefore assume that G is p–elementary.

Now if G is p–elementary, it is a product of a p–group and a cyclic group prime to p. Any irreducible complex representation of G is then induced from a linear representation on a subgroup [25, Thm.16].

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

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

It follows from Corollary 4.4 that the top composite map R_{F} (G) ⊗ Z_{(p)} → R_{Top}(G)) ⊗ Z_{(p)} is surjective.

The sum of the (ordinary) restriction maps to cyclic subgroups induces a rational injection on R_{F} (G) (see [27, 2.5,2.10]). Since R_{F} (G) is torsion–free, it follows again from Corollary 4.4 that the map R_{F} (G) ⊗ Z_{(p)} → R_{Top}(G)) ⊗ Z_{(p)} is injective. □

The proof of Theorem 4.1. Since an F –representation is free if and only if it is the sum of Galois conjugates of free G–representations, we can decompose R_{F} (G) as in Section 3, and conclude that RF^{free}(G_{odd}) = R^{free}(G_{odd}). It remains to show that R̃Top^{free}(G) is a torsion group with 2–primary exponent. For this we use the filtration of §3.

For R̃^{free}(G)/R̃h^{free}(G) this is easy since the k–invariant gives a homomorphism (via joins of free G–spheres) to (Z/2^{r})^{×} and this is a 2–group. The next quotient is also 2–primary torsion, by results of [30]: a sufficiently large 2–power join of a G–homotopy equivalence between two free G–spheres, which are linearly equivalent over G_{odd}, becomes s–normally cobordant to the identity. The point is that the normal invariant is detected by a finite number of 2–power congruences conditions among the Hirzebruch L–classes of the tangent bundles of the lens spaces, and this can be satisfied after sufficiently many joins.

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

Proposition 4.6. Let G = C(2^{r}q), q odd, and assume

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

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

Proof. We will take W = R_{−} ⊕ R_{−} ⊕ W_{0}, where W_{0} is the sum of all the irreducible 2– dimensional representations of G with isotropy of 2–power index. Note that the R_{−}-transfer is just the compact I_{−} transfer of one-sided codimension 1 surgery followed by adding rays to infinity, so whenever the I_{−}transfer is 0, the R_{−}-transfer will have to be 0. This was discussed in more detail in [I], Section 4. Step 1: If G has odd order, there is nothing to prove. Otherwise, let H ⊂ G be of index 2. If Res_{H}(σ) = 0, then trf_{R_{−}}(σ) ∈ Im(L_{1}^{h}(ZG^{−}) → L_{1}^{h}(ZH → ZG^{−})). But L_{1}^{h}(ZG^{−}) has exponent 2 [13, 12.3], so trf_{R_{−}}(2σ) = 0. Then take W = R_{−} ⊕ R_{−}. Note that this case applies to G = C(2q), so we can always get started. Step 2: We may assume that r ≧ 2. If Res_{H}(σ) ≠ 0 note that

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

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

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

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

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

From this we get

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

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

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

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

A similar argument to that in Step 2 above gives:

Proposition 4.7. If Res_{H}(trf_{W} (x)) = 0 for x ∈ L_{0}^{h}(ZG), then 4 · trf_{W ×R_{−}}(x) = 0.

Proof. Since

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

we conclude as above that both terms vanish. □

5. Excision in bounded surgery theory

A small additive category with involution A is a small additive category together with a contravariant endofunctor ∗ such that ∗^{2} = 1_{A}. Ranicki defines algebraic L–theory L_{∗}^{h}(A) for such categories and corresponding spectra 𝕃^{h}(A) with L_{∗}^{h}(A) = π_{∗}(𝕃^{h}(A)) [21]. The obstruction groups for bounded surgery are obtained this way for appropriately chosen additive categories. We shall also need a simple version of such groups. For this, the additive category must come equipped with a system of stable isomorphisms and a subgroup s ⊂ K_{1}(A), such that any composition resulting in an automorphism defines an element in s. The point here is that whenever two objects are stably isomorphic, there is a canonically chosen stable isomorphism, canonical up to automorphisms defining elements of s. In this situation Ranicki refines the definition of 𝕃^{h}(A) to give the simple L–theory spectrum 𝕃^{s}(A), by requiring appropriate isomorphisms to give elements of K_{1}(A) belonging to the subgroup s. More generally, we also get the 𝕃^{a}(A)-spectra for any involution invariant subgroup a with s ⊂ a ⊂ K_{1}(A), coinciding with 𝕃^{h}(A) when a = K_{1}(A).

Example 5.1. Let A be the category of free ZG-modules with a G-invariant Z-basis, and ZG-module morphisms. Two objects are stably isomorphic if and only if they have the same rank. The preferred isomorphisms are chosen to be the ones sending a Z-basis to a Z-basis, so automorphisms define elements of {±G} ⊂ K_{1}(ZG). In this situation one obtains Wall’s L^{s}-groups.

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

Example 5.2. If A is the category of free ZG-modules then A^{∧} is isomorphic to the category of projective ZG-modules and the L_{∗}^{p}(A) are Novikov’s original L^{p}-groups.

Suppose M is a metric space with the finite group G acting by isometries, R a ring with involution. In [10, 3.4] we defined an additive category G_{M,G} (R) with involution as follows:

Definition 5.3. An object A is a left R(G)-module together with a map f: A ⟶ F (M), where F (M) is the set of finite subsets of M, satisfying (i) f is G-equivariant. (ii) A_{x} = {a ∈ A | f(a) ⊆ {x}} is a finitely generated free sub R-module (iii) As an R-module A = ⨁_{x∈M} A_{x} (iv) f(a + b) ⊆ f(a) ∪ f(b)

(v) For each ball B ⊂ M, {x ∈ B | A_{x} ≠ 0} is finite. A morphism ϕ: A ⟶ B is a morphism of RG-modules, satisfying the following condition: there exists k so that the components ϕ_{n}^{m}: A_{m} ⟶ B_{n} (which are R-module morphisms) are zero when d(m, n) > k. Then G_{M,G} (R) is an additive category in an obvious way. The full subcategory of G_{M,G} (R), for which all the object modules are required to be free left RG-modules, is denoted C_{M,G}(R).

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

Example 5.4. Let ρ_{W} : G → O(W) be an orthogonal action of G on a finite dimensional real vector space W. We take M = W with the action through ρ_{W}, and orientation character det(ρ_{W}). This will be called the standard orientation on C_{W,G}(Z).

Remark 5.5. We will need to find a system of stable isomorphisms for the category C_{M,G}(R) to be able to do simple L–theory. To do this we choose a point x in each G-orbit, and an RG_{x}-basis for A_{x}, where G_{x} is the isotropy subgroup of x. We then extend that by equivariance to an R-basis of the module. Having a basis allows defining an isomorphism in the usual fashion. In each case we need to describe the indeterminacy in the choices coming from the choice of R-basis and points in the orbit. For our particular choices of M it will be easy to determine the subgroup s, so we will not formulate a general statement.

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

Definition 5.6. Let A be a full subcategory of an additive category U. Denote objects of A by the letters A through F and objects of U by the letters U through W. We say that U is A-filtered, if every object U has a family of decompositions {U = E_{α} ⊕ U_{α}}, so that (i) For each U, the decompositions form a filtered poset under the partial order that

E_{α} ⊕ U_{α} ≤ E_{β} ⊕ U_{β}, whenever U_{β} ⊆ U_{α} and E_{α} ⊆ E_{β}. (ii) Every map A → U, factors A → E_{α} → E_{α} ⊕ U_{α} = U for some α. (iii) Every map U → A factors U = E_{α} ⊕ U_{α} → E_{α} → A for some α. (iv) For each U, V the filtration on U ⊕ V is equivalent to the sum of filtrations {U =

E_{α} ⊕ U_{α}} and {V = F_{β} ⊕ V_{β}} i.e. to U ⊕ V = (E_{α} ⊕ F_{β}) ⊕ (U_{α} ⊕ V_{β})

The main excision results were proved in [18], [6], [7], [22]. We give a slight generalization of the L–theory results. Let 𝕂 denote the Quillen K–theory spectrum, and 𝕂^{−∞} its nonconnective delooping (with the K_{−i}-groups as homotopy groups).

Theorem 5.7. Let U be an A-filtered additive category with involution. Consider the map i: K_{0}(A^{∧}) → K_{0}(U^{∧}) induced by inclusion, and let k = i^{−1}(K_{0}(U)). There are fibrations of spectra

𝕂(A^{∧k}) → 𝕂(U) → 𝕂(U/A)

and

𝕂^{−∞}(A) → 𝕂^{−∞}(U) → 𝕂^{−∞}(U/A)

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

𝕃^{k}(A) → 𝕃^{h}(U) → 𝕃^{h}(U/A) .

More generally, if

(i) a ⊂ K_{i}(A),b ⊂ K_{i}(U^{∧}), and c ⊂ K_{i}((U/A)^{∧}), for i ≦ 1,

(ii) a = i^{−1}(b) and b → c is onto,

(iii) a, b, and c contain K_{0}(A), K_{0}(U) and K_{0}(U/A) respectively, if i = 0, and (iv) a, b and c contain the indeterminacy subgroup given by the system of stable isomor-

phisms in the case i = 1,

then we have a fibration of spectra

𝕃^{a}(A) → 𝕃^{b}(U) → 𝕃^{c}(U/A)

Proof. The K–theory statements are implicitly contained in [18]. A simpler, more modern proof and explicit statements are given in [6]. The first L–theory statement was proved in [7], and the other L–theory statements follow by the following argument: we have an exact sequence

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

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

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

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

a/K_{0}(A^{∧k}) ≅ b′/K_{0}(U),

so using the Ranicki–Rothenberg fibrations of spectra [22]

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

we get a fibration

𝕃^{a}(A) → 𝕃^{b′}(U) → 𝕃^{h}(U/A).

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

K_{1}(U/A)/ ker(∂) ≅ image(∂)

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

In Section 10 we need to use bounded surgery groups with geometric anti–structure generalizing the definition of [I], Section 4 (see [10]). The new ingredient is a counterpart to the automorphism θ: H → H at the metric space level.

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

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

Example 5.8. Bounded geometric anti–structures arise geometrically as above. The bounded R_{−} transfer sits in the long exact sequence

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

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

where w = det(ρ_{W}) is the standard orientation (see Example 5.4). The bounded LN–group

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

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

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

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

Definition 5.9. If A is an additive category, then the opposite category A^{op} is the category with the same objects as A but hom_{A^{op}}(A, B) = hom_{A}(B, A). The product category A × A^{op} is an additive category with involution given by ∗: (A, B) = (B, A) on objects and ∗: (α, β) = (β, α) on morphisms.

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

Lemma 5.10. Let b ⊆ K_{i}(A) for some i ≦ 1, and q = b × b ⊆ K_{i}(A × A^{op}). Then L_{n}^{q}(A × A^{op}) = 0 for all n.

Proof. Let P(A) denote the category with the same objects as A, but with morphisms given by A–isomorphisms. Then it suffices to prove that the quadratic category Q(A × A^{op}) ≃ P(A) via the hyperbolic map (see [29, p.122]). This shows that L_{∗}^{h}(A × A^{op}) = 0 and other decorations follow trivially from the Ranicki Rothenberg exact sequences (note that the Tate cohomology H^{∗}(q) = 0). The result for lower L–groups follows by replacing A by C_{R}(A).

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

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

and the right hand side is a hyperbolic form. □

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

Example 5.11. Let M = M_{1} ∪ M_{2} be a metric space given as the union of two sub–metric spaces M_{1} and M_{2}, where we denote M_{1} ∩ M_{2} by N. Suppose that

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

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

d(x, N) > l, then d(x, M_{2}) > k (resp. d(x, M_{1}) > k).

The category C_{M,H}(R) has a bounded geometric antistructure (α, u) given by (θ_{H}, θ_{M}, b, w) as in Example 5.8, with θ_{M}(m) = t · m and θ_{H}(h) = tht^{−1} for a fixed t ∈ G − H. Next, observe that the category

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

because of our separation condition (iii). Moreover, the functor T: C_{M_{1},H}^{>N}(R) → C_{M_{2},H}^{>N}(R)^{op} defined by T (A, f) = (A^{∗}, θ_{M}^{−1}∘f^{∗}) on objects and T (ϕ) = ϕ^{∗} on morphisms is an equivalence of categories. We are thus in the A×A^{op} situation described above and L_{∗}^{h}(C_{M,H}^{>N}(R), α, u) = 0 by Lemma 5.10.

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

Theorem 5.12. Suppose that (C_{M,H}(R), α, u) has a bounded geometric antistructure (α, u) given by (θ_{H}, θ_{M}, b, w), such that:

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

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

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

Corollary 5.13. Let G = C(2^{r}q), q odd, be a cyclic group and H ⊂ G the subgroup of index 2. Let W be a G–representation, and N = ⋃{W^{K} | [G : K] is odd}. Then L_{n}^{I}(C_{W,H}^{>N} (Z), α, u)(q) = 0 on the top component, where (α, u) is the antistructure given above.

The proof of Theorem 5.12. We extend the given H–action on M to a simplicial action of G = ⟨H, t | t^{−1}ht = θ_{H}(h), t^{2} = b⟩ as described above. The proof is by induction on cells, so suppose that K is obtained from L by attaching exactly one G–equivariant k– cell D^{k} × G/G_{0}. Since M_{(H,θ)} ⊂ O(L) = N, it follows that G_{0} ⊂ H and we may write G/G_{0} = H/G_{0} ⊔ tH/G_{0}. Now we define M_{1} = O(L ∪ (D^{k} × H/G_{0})) and consider the category A = C_{M_{1},H}^{>O(L)}(R). By construction, we have

C_{M,H}^{>N}(R) = A × A^{op}

which has trivial L–theory by Lemma 5.10. Since the Tate cohomology of the K_{1} decoration I vanishes, we get L_{n}^{I}(C_{M,H}^{>N} (R), α, u) = 0. □

6. Calculations in bounded K-theory

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

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

W = W [0] ⊕ W [1] ⊕ ⋯ ⊕ W [r]

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

We study bounded K–theory by means of equivariant filtrations of the control space. The basic sequence is (see [10]):

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

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

Recall that t denotes a generator of G, and thus acts as an isometry on the control spaces M we use in the bounded categories C_{M,G}(Z). Let t_{∗} denote the action of t on bounded K–theory induced by its action on the control space.

Lemma 6.1. Let W = W_{1} ⊕ W_{2}, where W_{1} is a complex 2–dimensional sub-representation of W with minimal isotropy subgroup K ≠ G. Then

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

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

Proof. The bounded category C_{⋃ ℓ_{α}×W_{2},G}^{>W_{2}}(Z) of germs away from W_{2} has effective fundamental group K, as defined in [10, 3.13]. It therefore has the same K–theory as C_{R^{k+1}}(C_{pt}(ZK)). The other case is similar. The identification of ∂_{i+1} with 1 − t_{∗} is discussed in detail in the proof of Proposition 6.7. □

Since K_{−j}(ZK) = 0 for j ≧ 2 by [8], this Lemma gives vanishing results for bounded K–theory as well.

Lemma 6.2. Suppose that W is complex, and W^{G} = 0. Then the inclusion map induces an isomorphism

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

for i ≦ 1.

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

Then applying the first part of Lemma 6.1, we get the calculations

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

and

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

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

[8], and K_{i}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×W′′×R_{−}}(Z)) = 0 for i ≦ 3. From the filtration sequence, it follows that

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

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

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

Corollary 6.5. K_{i}(C_{W ×R_{−},G}^{>W_{max}×R_{−}}(Z)) = 0 for i ≦ 2.

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

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

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

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

Lemma 6.6. Let A be an additive category (with involution). Then the map R → R sending x to −x induces minus the identity on K-theory (and L-theory) of C_{R}(A).

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

Consider the diagram

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

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

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

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

As above, t_{∗} denotes the induced action of a generator t ∈ G on K–theory, and ε: G → {±1} the non-trivial action of G on R_{−}.

Proposition 6.7. Let W be a complex 2-dimensional G–representation with W^{G} = 0, and isotropy subgroup K. Then under the isomorphisms of (6.3) and (6.4) the complex

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

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

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

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

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

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

K_{i+1}(C_{C×R,K}^{>∂C×R}(Z)),

where C is the region between two adjacent half–lines of ⋃ ℓ_{α}. This follows since the regions in the complement of ⋃ ℓ_{α} × R_{−} are disjoint and the boundedness condition ensures there is no interference. Similarly the isomorphism of the range of ∂_{a} with K_{i−2}(Z) is obtained by noticing that every element is induced from C_{h×R,K}^{>R} where h is just one half–line in ⋃ ℓ_{α}, and we can think of ∂C = h ∪ th. To compute the boundary we first take the standard boundary to C_{∂C×R,K}(Z) which is an isomorphism, and then map away from 0 × R. It follows from the proof of Lemma 6.6 above that this map is of the form a ↦ (a, −a) in K or L–theory. In this picture, the support of one of the boundary components is along h and the other along th. We need to use the group action to associate both elements to the same ray. Since t flips the R–factor, we get a change of sign before adding, so t_{∗} = ε(t) = −1 and ∂_{a} sends a to 2a.

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

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

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

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

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

Lemma 6.8. Let W be a complex 2-dimensional G–representation with proper isotropy group K. The boundary map

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

is zero for i ≦ 1.

Proof. If i ≦ −1 the domain of this boundary map is zero, so the result is trivial. For i = 1 we use the injection Wh(C_{R_{−},G}(Z)) → Wh(C_{R_{−},G}(Ẑ_{2})), which follows from the vanishing of SK_{1}(ZG). But Wh(C_{R_{−},G}(Ẑ_{2})) = Wh(Ẑ_{2}G)/ Wh(Ẑ_{2}H) and the group Wh(Ẑ_{2}G) = ∏_{d|q} Ẑ_{2}[ζ_{d}]G_{2}, where G_{2} ⊂ G is the 2-Sylow subgroup and q is the odd part of the order of G. Now Oliver [17, Thm. 6.6] constructed a short exact sequence

1 → Wh(Ẑ_{2}[ζ_{d}]G_{2}) → Ẑ_{2}[ζ_{d}]G_{2} → ⟨−1⟩ × G_{2} → 1

by means of the integral 2-adic logarithm. This sequence is natural with respect to inclusion of subgroups, so we may use it to compare Wh(Ẑ_{2}G) and Wh(Ẑ_{2}H). Since each corresponding term injects, and the middle quotient is Z-torsion free, we conclude that Wh(C_{R_{−},G}(Ẑ_{2})) is also Z-torsion free. Since K_{2}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z) = K_{0}(ZK) is torsion (except for Z = K_{0}(Z) which is detected by projection to the trivial group), the given boundary map is zero.

For i = 0, we use the surjection K_{0}(Q̂K) → K_{−1}(ZK), and compute with Q̂ coefficients and i = 1. We will list the steps, and leave the details to the reader. First, compute that K_{1}(C_{R_{−},G}(Q̂)) = K_{1}(Q̂G)/K_{1}(Q̂H) surjects onto K_{1}(C_{⋃ ℓ_{α}×R_{−},G}(Q̂)), by means of a braid containing the cone point inclusions into C_{R_{−},G}(Q̂) and C_{⋃ ℓ_{α}×R_{−},G}(Q̂). Second, prove that K_{1}(C_{⋃ ℓ_{α}×R_{−},G}(Q̂)) fits into a short exact sequence

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

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

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

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

Corollary 6.9. Let W be a complex G–representation with W^{G} = 0. Then the inclusion induces an isomorphism K_{i}(C_{R_{−},G}(Z)) → K_{i}(C_{W ×R_{−},G}(Z)) for i ≦ 0, and an injection for i = 1. If K_{−1}(ZK) = 0 for the maximal proper isotropy groups K of W, then K_{1}(C_{R_{−},G}(Z)) ≅ K_{1}(C_{W ×R_{−},G}(Z)).

Proof. We may assume that dim W = 2, and apply the filtering argument again. By (6.3) and (6.4) we get K_{i}(C_{W ×R_{−},G}^{>R_{−}}(Z)) = 0 for i ≦ 0, and K_{1}(C_{W ×R_{−},G}^{>R_{−}}(Z)) is a quotient of

K_{1}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) = K_{−1}(ZK). Since the composition

K_{1}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) → K_{1}(C_{W ×R_{−},G}^{>R_{−}}(Z)) → K_{0}(C_{R_{−},G}(Z))

is zero by Lemma 6.8, the result follows for i ≦ 0.

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

boundary map K_{2}(C_{W ×R_{−},G}^{>⋃ ℓ_{α}×R_{−}}(Z)) = K_{−1}(ZK) to K_{1}(C_{⋃ ℓ_{α}×R_{−},G}^{>R_{−}}(Z)) = K_{−1}(ZK) is multiplication by 2, and hence injective. Then we make the same argument, using Lemma 6.8.

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

K_{1}(C_{R_{−},G}(Z)) ⟶^{≈} K_{1}(C_{W ×R_{−},G}(Z)). □

7. The double coboundary

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

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

(see [13]) has an algebraic description by means of a “double coboundary” homomorphism

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

In this section, we will give a brief description due to Ranicki [19] of this homomorphism (see also [20, §6.2] for related material on “interlocking” exact sequences in K and L–theory).

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

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

by sending g: S^{n} → X to

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

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

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

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

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

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

which can be spliced together from the short exact sequences

\begin{matrix}0 → π_{n}(X)/I_{n} → π_{n}(Y) → J_{n} → 0 \\ 0 → J_{n} → π_{n}(f) → I_{n−1} → 0 . \end{matrix} ; (7.1)

Then it follows directly from the definitions that the double coboundary

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

from the Tate cohomology sequences induced by (7.1) is given by the composite

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

If we can pick the map f: X → Y appropriately, say with I_{n} = 0 and I_{n−1} = π_{n−1}(X), this gives an algebraic description of Δ.

In later sections we will use the relative Tate cohomology groups H^{i}(Δ), which are just (by definition) the relative Tate cohomology groups [20, p.166] of the map π_{n}(Y) → π_{n}(f) in the long exact sequence above. These groups fit into the commutative braids given in [19] which will be used in the proofs of Theorems A-C.

We now give some examples, with G denoting a finite cyclic group as usual. These arise from homotopy groups of certain fibrations of algebraic K-theory spectra.

Example 7.2. There is an exact sequence [13]

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

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

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

The point here is that ker(Wh(ZG) → Wh(Ẑ_{2}G)) = 0 [17], and the map K̃_{0}(ZG) → K̃_{0}(Ẑ_{2}G) is zero by a result of Swan [26]. We could also use the exact sequence

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

to compute the same map Δ.

Example 7.3. There is an exact sequence

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

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

where Wh(C_{R_{−},G}(Z)) = Wh(ZH → ZG) and K̃_{0}(C_{R_{−},G}(Z)) = K̃_{0}(ZH → ZG). The injectivity on the left follows because K_{2}(C_{R_{−},G}(Q̂)) is a quotient of K_{2}(Q̂G), mapping trivially through K_{1}(ZG) into Wh(C_{R_{−},G}(Z)) (since SK_{1}(ZG) = 0 [17]). We therefore get an algebraic description of δ^{2} : H^{i}(K̃_{0}(ZH → ZG^{−})) → H^{i}(Wh(ZH → ZG^{−})) as used in the statement of Theorem B.

Example 7.4. There is an exact sequence

(7.4) 0 → Wh(C_{W ×R_{−},G}(Z)) → Wh(C_{W ×R_{−},G}(Ẑ ⊕ Q)) →

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

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

Lemma 7.5. For complex G–representations W_{1} ⊆ W with W^{G} = 0, there is a commutative diagram

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

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

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

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

Proposition 7.6. Let G_{1} < G be a subgroup of odd index, and H_{1} < G_{1} have index 2, then there are twisted restriction maps

H^{i}(K̃_{0}(ZH →ZG^{−})) ⟶^{Res} H^{i}(K̃_{0}(ZH_{1}→ZG_{1}^{−}))

and

H^{i}(Wh(ZH →ZG^{−})) ⟶^{Res} H^{i}(Wh(ZH_{1}→ZG_{1}^{−}))

such that the diagram

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

commutes.

Proof. The vertical maps are twisted restriction maps given by composing the twisting isomorphisms H^{i}(K̃_{0}(ZH → ZG^{−})) ≅ H^{i+1}(K̃_{0}(ZH → ZG)) and H^{i}(Wh(ZH → ZG^{−}) ≅ H^{i+1}(Wh(ZH →ZG), discussed in [I], Section 4, with the restriction maps induced by the inclusion ZG_{1} → ZG of rings with involution. Since G_{1} < G has odd index, H_{1} = H∩G_{1} and the composition Res_{G_{1}} ∘ Ind_{H} lands in the image of Ind_{H_{1}} by the double coset formula. □

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

Proposition 7.7. Let G_{1} < G be an odd index subgroup, and H_{1} < G_{1} have index 2. Suppose that W only has proper isotropy subgroups of 2–power index. Then there are a twisted restriction maps H^{i}(K̃_{0}(C_{W ×R_{−},G}(Z))) → H^{i}(K̃_{0}(C_{Res W ×R_{−},G_{1}}(Z))) and H^{i}(Wh(C_{W ×R_{−},G}(Z))) → H^{i}(Wh(C_{Res W ×R_{−},G_{1}}(Z))) which commute with the corresponding double coboundary maps δ_{W}^{2} and δ_{Res W}^{2} .

8. Calculations in bounded L–theory

Suppose that σ = σ(f) ∈ L_{0}^{h}(ZG) is the surgery obstruction arising from a normal cobordism between S(V_{1}) and S(V_{2}), as in the statement of [I], Theorem 3.5. In this section, we establish two important properties of trf_{W} (σ) in preparation for the proof of Theorem C. Unless otherwise mentioned, all bounded categories will have the standard orientation (see Example 5.4).

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

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

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

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

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

Theorem 8.1. Suppose W is a complex G–representation with no R_{+} summands. If trf_{W ×R_{−}}(σ) ∈ L_{2k+1}^{h}(C_{W ×R_{−},G}(Z)) is a torsion element, then

(i) there exists a torsion element σ̂ ∈ L_{2k+1}^{h}(C_{R_{−},G}(Z)) such that

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

(ii) there exists a torsion element σ̂ ∈ L_{2k+1}^{p}(ZG, w) such that

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

where dim W = 2k.

We remark that the condition “trf_{W ×R_{−}}(σ) is a torsion element” follows from the assumption Res_{H}(V_{1} ⊕ W) ⊕ R_{+} ∼_{t} Res_{H}(V_{2} ⊕ W) ⊕ R_{+} in Theorem C, as an immediate consequence of Proposition 4.7. Before giving the proof, we need some preliminary results. When the L–theory decoration is not explicitly given, we mean L^{⟨−∞⟩}.

Lemma 8.2. Let W be a complex G–representation. Then

L_{2k+1}(C_{W,G}(Z)) ⊗ Q = 0

for k ≧ 0

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

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

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

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

Then

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

which is torsion for n even, and

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

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

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

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

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

which may be identified (using the isomorphisms above and Proposition 6.7) with a geometrically induced map

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

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

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

\xymatrix{L_{2k+1}(C_{⋃ ℓ_{α}×W′′,G}^{>W′′} (Z)) \ar[r]^{≈} & L_{2k+1}(C_{W,G}^{>W′′} (Z))}

is a rational isomorphism and so

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

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

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

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

and find that

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

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

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

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

Another computation we will need is

Lemma 8.3. L_{n}(C_{W ×R_{−},G}^{>R_{−}}(Z)) ⊗ Q = 0 for W a complex G-representation.

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

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

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

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

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

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

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

and

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

Then we need the L–group sequence

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

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

L_{n}(D_{0}) = L_{n−2}(C_{W′′,K}(Z))

and

L_{n}(D_{1}) = L_{n−3}(C_{W′′,K}(Z)).

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

L_{n}(D_{0}) = L_{n−2−|W′′|}(ZK)

and

L_{n}(D_{1}) = L_{n−3−|W′′|}(ZK).

Now the boundary map

(8.6)  L_{n}(D_{1}) → L_{n−1}(D_{0})

in the sequence is just multiplication by 1 − w(t) = 2 (w(t) = −1 since R_{−} is non-oriented), and this is a rational isomorphism. Therefore, L_{n}(A/B) = 0 by (8.5) and substituting back into (8.4), the L_{n}(A) = 0.

□

Now a more precise result in a special case:

Lemma 8.7. Let W be a complex 2–dimensional G-representation with W^{G} = 0. Then L_{3}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)) = 0 Proof. We have an exact sequence

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

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

arising from the orbit type filtration. Here

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

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

= ker(K_{−1}(ZK) ⟶^{2} K_{−1}(ZK))

by Proposition 6.7, where K is the isotropy subgroup of W. But K_{−1}(ZK) is torsion–free, so I = {0} ⊆ K_{2}. Now substitute the computations

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

into the exact sequence. The boundary map

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

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

Remark 8.8. The same method shows that L_{1}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)) = L_{3}^{p}(ZK) = Z/2 for dim W = 2 as above, assuming that K ≠ 1.

Our final preliminary result is a Mayer–Vietoris sequence:

Lemma 8.9. Let W be a complex G–representation with W^{G} = 0. Let W = W_{1} ⊕ W_{2} be a direct sum decomposition, where W_{1} = W_{max}.

(i) There is a long exact sequence

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

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

of bounded L-groups, where

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

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

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

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

L_{n}(C_{M,G}^{>U}(Z)) ⊕ L_{n}(C_{M,G}^{>V}(Z)) → L_{n}(C_{M,G}^{>U∪V} (Z) where U, V are nice G–invariant subspaces of the control space M, follows from an excision isomorphism

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

and standard diagram-chasing. We apply this to M = W × R_{−}, U = W_{1} × R_{−} and V = W_{2} × R_{−}, where U ∩ V = R_{−}. The decorations follow from Section 5 and Corollary 6.5.

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

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

and an excision isomorphism

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

But L_{3}^{s}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z)) = 0 by Lemma 8.7. □

Corollary 8.10. Suppose W is a complex G–representation with no R_{+} summands. Then the image of trf_{W ×R_{−}}(σ) is zero in L_{2k+1}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)), where dim W = 2k.

Proof. First recall from [I], Theorem 3.6 that trf_{W ×R_{−}}(σ) ∈ L_{2k+1}^{c_{1}}(C_{W ×R_{−},G}(Z)) where c_{1} = Im(Wh(ZG) → Wh(C_{W ×R_{−},G}(Z))) under the cone point inclusion. Therefore, the image of trf_{W ×R_{−}}(σ) “away from R_{−}” lands in L_{2k+1}^{s}(C_{W ×R_{−},G}^{>R_{−}}(Z)). We may therefore apply the Mayer–Vietoris sequence from Lemma 8.9.

As usual, we may assume that W has only 2–power isotropy subgroups, and we will argue by induction on dim W and on the orbit type filtration of W. The result is true for dim W = 2 (by Lemma 8.7) or for W a free representation (by a direct calculation following the method of Lemma 8.3). For sets of 2–power index subgroups (ordered by inclusion, with repetitions allowed), we say that {K′_{1}, …, K′_{s}} < {K_{1}, …, K_{t}} if s < t or s = t, K′_{i} = K_{i} for 1 ≦ i < j ≦ s, and K′_{j} ⫋ K_{j}. Let Iso(W) = {K_{1}, …, K_{t}} denote the set of isotropy subgroups of W ordered by inclusion. Define an ordering by W′ < W if (i) dim W′ < dim W or (ii) dim W′ = dim W but Iso(W′) < Iso(W).

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

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

from Lemma 8.9, where W_{2} < W and W_{1} < W. If dim W ≡ 0 mod 4, we get an injection

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

again from Lemma 8.9, where U is any 2–dimensional free G–representation. Notice that W_{2} ⊕ U < W in the ordering above. Consider the following commutative diagram (with dim W ≡ 2 mod 4):

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

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

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

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

The proof of Theorem 8.1. Part (i) follows from Corollary 8.10: we know that the image of our obstruction trf_{W ×R_{−}}(σ) is zero “away from R_{−}”, so comes from a torsion element in σ̂ ∈ L_{2k+1}^{h}(C_{R_{−},G}(Z)) (by Lemma 8.3, L_{2k+2}(C_{W ×R_{−},G}^{>R_{−}}(Z)) is torsion).

For part (ii) observe that L_{2k+1}^{h}(C_{R_{−},G}^{>0}(Z)) = L_{2k}^{p}(ZH) is torsion–free (except for the Arf invariant Z/2 which is detected by the trivial group, so does not matter). Hence the image of trf_{W ×R_{−}}(σ) vanishes in L_{2k+1}^{h}(C_{W ×R_{−},G}^{>0}(Z)). The exact sequence

⋯ → L2k+1^{U}(ZG, w) → L_{2k+1}^{h}(C_{W ×R_{−},G}(Z)) → L_{2k+1}^{h}(C_{W ×R_{−},G}^{>0}(Z)),

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

σ̂ ∈ Im(L2k+1^{U}(ZG, w) → L_{2k+1}^{p}(ZG, w))

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

Remark 8.11. In Theorem 10.7 we use the sharper result σ̂ ∈ L2k+1^{U}(ZG, w) to improve our unstable similarity results by removing extra R_{−} factors. The L-group decoration U is independent of W, since K_{0}(C_{R_{−},G}(Z)) → K_{0}(C_{W ×R_{−},G}(Z) is injective by Corollary 6.9, and hence U = Im(K̃_{0}(ZH) → K̃_{0}(ZG)).

9. The proof of Theorem C

By replacing V_{i} by V_{i} ⊕ U if necessary, where U is free and 2–dimensional, we may assume that dim V_{i} ⊕ W ≡ 0 mod 4. This uses the s–normal cobordism condition. By the top component argument (see [I], Theorem 3.8) and Proposition 7.7, we may also assume that W has only 2–power isotropy. We have a commutative diagram analogous to [I] (8.1).

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

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

This diagram for W can be compared with the one for W_{1} = W_{max} via the inclusion maps, and we see that the K-theory terms map isomorphically by Lemma 6.2. Now by Theorem 8.1 there exists a torsion element

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

which hits trf_{W ×R_{−}} under the cone point inclusion. In particular, σ̂ vanishes “away from R_{−}”. The main step in the proof of Theorem C is to show that the torsion subgroup of L_{1}^{p}(C_{W_{1}×R_{−},G}(Z)) essentially injects into the relative group H^{1}(Δ_{W_{1}×R_{−}}) of the double coboundary.

We have the exact sequence

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

and Y = ker(Wh(C_{R_{−},G}(Z)) → Wh(C_{W_{1}×R_{−},G}(Z)) is zero, by Corollary 6.9. But the first term L_{2}^{s}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z)) is a torsion group, by Lemma 8.3, and the next term in the sequence L1^{Y}(C_{R_{−},G}(Z)) = L_{1}^{s}(C_{R_{−},G}(Z)) is torsion–free by [I], Lemma 7.1 so the middle map L1^{Y}(C_{R_{−},G}(Z)) ↣ L_{1}^{s}(C_{W_{1}×R_{−},G}(Z)) is an injection.

Since L_{1}^{s}(C_{W_{1}×R_{−},G}^{>R_{−}}(Z)) = Z/2 by Remark 8.8 we conclude that the previous group

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

By substituting this computation into the exact sequence

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

of (9.1), we see that the torsion subgroup of

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

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

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

This completes the proof of Theorem C.

A very similar argument can be used to give an inductive criterion for non–linear similarity without the R_{+} summand (generalizing Theorem A). In the statement we will use the analogue to k = ker(K̃_{0}(ZH) → K̃_{0}(ZG)), namely

k_{W} = ker(Wh(C_{W ×R_{−},G}^{>0}(Z)) → K̃_{0}(ZG)).

By Corollary 6.9

ker(Wh(ZG) → Wh(C_{W ×R_{−},G}(Z)) ≅ Im(Wh(ZH) → Wh(ZG)) ≅ Wh(ZH)

so we have a short exact sequence

0 → Wh(ZG)/ Wh(ZH) → Wh(C_{W ×R_{−},G}(Z)) → k_{W} → 0 .

It follows from our K–theory calculations that k_{W} = k_{W_{max}} and that k_{W} = k whenever K_{−1}(ZK) = 0 for all K ∈ Iso(W).

Theorem 9.2. Let V_{1} = t^{a_{1}} +⋯+t^{a_{k}} and V_{2} = t^{b_{1}} +⋯+t^{b_{k}} be free G–representations. Let W be a complex G–representation with no R_{+} summands. Then there exists a topological similarity V_{1} ⊕ W ⊕ R_{−} ∼_{t} V_{2} ⊕ W ⊕ R_{−} if and only if

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

δ: H^{0}(k_{W_{max}}) → H^{1}(Wh(ZG^{−})/ Wh(ZH)),

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

Proof. In this case, we have a torsion element σ̂ ∈ L_{1}^{h}(C_{W_{1}×R_{−},G}(Z)) which maps to our surgery obstruction trf_{W ×R_{−}} under the cone point inclusion. As in the proof of Theorem C, the torsion subgroup of L_{1}^{h}(C_{W_{1}×R_{−},G}(Z)) injects into H^{1}(Wh(C_{W_{1}×R_{−},G}(Z))) ≅ H^{1}(Wh(C_{W_{1}×R_{−},G}(Z))). Therefore σ̂ ∈ L_{1}^{h}(C_{W_{1}×R_{−},G}(Z)) vanishes if and only if the element {Δ(V_{1})/Δ(V_{2})} is in the image of the coboundary

δ: H^{0}(k_{W_{max}}) → H^{1}(Wh(ZG^{−})/ Wh(ZH))

as required. □

10. Iterated R_{−} transfers

We now apply Theorem 5.12 to show that iterated R_{−} transfers do not lead to any new similarities.

Theorem 10.1. Suppose that W is a complex G–representation with no R_{+} summands. Then V_{1} ⊕ W ⊕ R_{−}^{l} ⊕ R_{+} ∼_{t} V_{2} ⊕ W ⊕ R_{−}^{l} ⊕ R_{+} for l ≧ 1 implies V_{1} ⊕ W ⊕ R_{−} ⊕ R_{+} ∼_{t} V_{2} ⊕ W ⊕ R_{−} ⊕ R_{+}.

The first step in the proof is to show injectivity of certain transfer maps. For any homomorphism w: G → {±1}, we will use the notation (C_{W,G}(Z), w) to denote the antistructure where the involution is g ↦ w(g)g^{−1} at the cone point. The standard orientation (5.4) has w = det(ρ_{W}), but we will need others in this section. Let ϕ = det(ρ_{R_{−}}) for short, and notice that ϕ: G → {±1} is non–trivial.

Lemma 10.2. The transfer map trf_{R_{−}}: L_{2k+1}^{p}(ZG, w) → L_{2k+2}^{p}(C_{R_{−},G}(Z), wϕ) is injective, where w is the non–trivial orientation.

Proof. Since L_{2k+2}^{p}(C_{R_{−},G}(Z), w) ≅ L_{2k+2}^{p}(ZH →ZG), we just have to check that the previous map in the L^{p} version of (5.8) is zero (using the tables in [13]). □

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

Proposition 10.3. The bounded transfer

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

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

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

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

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

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

The diagram Φ of groups (as in [28, Chap.11]) contains Γ_{H} ≅ Z × H as the fundamental group of S(R_{−} ⊕ R_{−})/G (see [11, 5.9]).

The groups LNS_{∗}(Φ) have a geometric bordism description in terms of triples L^{n−2} ⊂ N^{n−1} ⊂ M^{n} where each manifold has fundamental group G, N is characteristic in M, and L is characteristic in N, with respect to the index two inclusion H < G. There is also an algebraic description for LNS_{∗}(Φ) in terms of “twisted antistructures” [12] as for the other groups LN(ZH →ZG, wϕ) [28, 12C] and for LS(Φ, w) [20, 7.8.12]. Substituting these descriptions into the braid gives:

(10.5)

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

The antistructure (ZG, α, u) is the twisted antistructure obtained by scaling with an element a ∈ G − H. The antistructure (ZΓ_{H}, α̃, ũ) is the one defined by Ranicki [20, p.805], then scaled by ã ∈ Γ_{H}, where ã maps to a under the projection Γ_{H} → G. Since Γ_{H} ≅ Z × H, we have an exact sequence (see [16, Theorem 4.1])

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

and it follows that L_{n}^{p}(ZH → ZΓ_{H}, α̃, ũ) ≅ Ln−1^{⟨−1⟩}(ZH, α, u). It is not difficult to see that L2k^{⟨−1⟩}(ZH, α, u) is torsion–free (except for the Arf invariant summand) by a similar argument to [I], Theorem 5.2, using the L^{p} to L^{⟨−1⟩} Ranicki-Rothenberg sequence. We first check that L_{2k}^{p}(ZH, α, u) is torsion–free (again except for the Arf invariant summand) from the tables in [13, 14.21].

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

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

The transfer trf_{R_{−}}: L_{2k+1}^{p}(ZG, w) → L_{2k+2}^{p}(ZH → ZG) is injective by Lemma 10.2, and since w(a) = −1, the map

L2k^{⟨−1⟩}(ZH, α, u) ⟶^{1−w(a)} L_{2k}^{p}(ZH, α, u)

is also injective (except for the Arf invariant summand). Therefore L_{2k+1}^{p}(ZG, w) must inject into the relative group L_{2k+1}^{p}(ZΓ_{H} →ZG, α̃, ũ) ≅ L_{2k+3}^{p}(C_{R_{−}^{2},G}(Z), w). □

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

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

for any G-representation V with no R_{+} summands.

Lemma 10.6. This is a pull–back square of L^{⟨−∞⟩} spectra.

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

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

and 𝕃ℕ(C_{V ×R_{−},G}(Z), wϕ) ≅ 𝕃(C_{V,H}(Z), α, u) respectively by (5.8), and the fibre of the cone point inclusion

𝕃(C_{pt}(ZH), α, u) → 𝕃(C_{V,H}(Z), α, u)

is 𝕃(C_{V,H}^{>0}(Z), α, u). But Theorem 5.12 shows that 𝕃(C_{V,H}^{>0}(Z), α, u) is contractible, and therefore the cone point inclusion is a homotopy equivalence. It follows that the fibres of the horizontal maps are also homotopy equivalent. □

The proof of Theorem 10.1. It is enough to prove that trf_{W ×R_{−}^{l+1}}(σ) = 0, for l ≧ 1, implies

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

The pullback squares provided by Lemma 10.6 can be combined as follows. Consider the diagram of L^{⟨−∞⟩}-spectra

\xymatrix{𝕃(C_{pt}(ZG), w) \ar[r]^{c_{∗}} \ar[d]_{trf_{R_{−}}} & 𝕃(C_{R_{−},G}(Z), wϕ) \ar[r]^{c_{∗}} \ar[d]_{trf_{R_{−}}} & 𝕃(C_{W ×R_{−}^{2},G}(Z), w) \ar[d]_{trf_{R_{−}}} \\ 𝕃(C_{R_{−},G}(Z), wϕ) \ar[r]^{c_{∗}} & 𝕃(C_{R_{−}^{2},G}(Z), w) \ar[r]^{c_{∗}} & 𝕃(C_{W ×R_{−}^{3},G}(Z), wϕ)}

whose outer square (V = W × R_{−}^{2}) and the left-hand square (V = R_{−}) are both pullback squares, and hence so is the right-hand square. Next consider the diagram of L^{⟨−∞⟩}-spectra

\xymatrix{𝕃(C_{pt}(ZG), w) \ar[r]^{c_{∗}} \ar[d]_{trf_{R_{−}}} & 𝕃(C_{W ×R_{−},G}(Z), wϕ) \ar[d]_{trf_{R_{−}}} \\ 𝕃(C_{R_{−},G}(Z), wϕ) \ar[r]^{c_{∗}} \ar[d]_{trf_{R_{−}}} & 𝕃(C_{W ×R_{−}^{2},G}(Z), w) \ar[d]_{trf_{R_{−}}} \\ 𝕃(C_{R_{−}^{2},G}(Z), w) \ar[r]^{c_{∗}} & 𝕃(C_{W ×R_{−}^{3},G}(Z), wϕ)}

The lower square was just shown to be a pull-back, and the upper square is another special case of Lemma 10.6 (with V = W × R_{−}). Therefore the outer square is a pull-back, and this is the one used for the case l = 2.

Now we apply homotopy groups to these pull-back squares (using the fact that L^{⟨−∞⟩} ≡ L^{⟨−1⟩} to obtain the lower squares in the commutative diagram:

\xymatrix{L_{2k+2}^{p}(C_{W ×R_{−},G}^{>0}(Z)) \ar[r] \ar[d]^{≅} & L_{2k+1}^{p}(ZG, w) \ar[r]^{c_{∗}} \ar[d] & L_{2k+1}^{p}(C_{W ×R_{−},G}(Z), w) \ar[d] \\ L2k+2^{⟨−1⟩}(C_{W ×R_{−},G}^{>0}(Z)) \ar[r] \ar[d]^{≅} & L2k+1^{⟨−1⟩}(ZG, w) \ar[r]^{c_{∗}} \ar[d]_{trf_{R^{2}_{−}}} & L2k+1^{⟨−1⟩}(C_{W ×R_{−},G}(Z), w) \ar[d]^{trf_{R^{2}_{−}}} \\ L2k+4^{⟨−1⟩}(C_{W ×R_{−}^{3},G}^{>0}(Z)) \ar[r] & L2k+3^{⟨−1⟩}(C_{R_{−}^{2},G}(Z), w) \ar[r]^{c_{∗}} & L2k+3^{⟨−1⟩}(C_{W ×R_{−}^{3},G}(Z), w)}

The vertical maps in the top row squares are induced by the change of K-theory decoration. We need some information about the maps in this diagram.

(i) The upper left-hand vertical map is an isomorphism, since K_{−1}(C_{W ×R_{−},G}^{>0}(Z)) = 0. (ii) The lower left-hand vertical map is an isomorphism, by Lemma 10.6. (iii) The transfer map

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

is injective, from Proposition 10.3. (iv) The map

L_{2k+3}^{p}(C_{R_{−}^{2},G}(Z), w) → L2k+3^{⟨−1⟩}(C_{R_{−}^{2},G}(Z), w)

is injective, by [I], Corollary 6.13.

(v) The composite of the middle two vertical maps in the diagram is injective, by combining parts (iii)-(iv).

Suppose that trf_{W ×R_{−}^{3}}(σ) = 0. We have proved that the transfer map

trf_{R_{−}^{2}} : L_{2k+1}^{p}(ZG, w) → L2k+3^{⟨−1⟩}(C_{R_{−}^{2},G}(Z), w)

is injective in part (v). Since trf_{W ×R_{−}}(σ) = c_{∗}(σ̂) for some σ̂ ∈ L_{2k+1}^{p}(ZG, w), it follows by a diagram chase that trf_{W ×R_{−}}(σ) = 0 and we are done. □

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

Theorem 10.7. Suppose that W is a complex G–representation with W^{G} = 0. Then V_{1} ⊕ W ⊕ R_{−}^{l} ∼_{t} V_{2} ⊕ W ⊕ R_{−}^{l} for l ≧ 3 implies V_{1} ⊕ W ⊕ R_{−}^{3} ∼_{t} V_{2} ⊕ W ⊕ R_{−}^{3}.

Proof. This follows from a similar argument, using Theorem 9.2 instead of Theorem C. The injectivity results of Lemma 10.2 and Proposition 10.3 also hold for trf_{1} and trf_{2} on L2k+1^{U}(ZG, w), where U = Im(K̃_{0}(ZH) → K̃_{0}(ZG)) as in Remark 8.11. This L-group decoration fits in exact sequences with L_{n}^{p}(ZH) so the previous calculations for injectivity apply again. The details are left to the reader. □

11. The proof of Corollary 2.4

Suppose that G = C(4q), q odd, and that V_{1}⊕R_{−}⊕R_{+} ∼_{t} V_{2}⊕R_{−}⊕R_{+} with dim V_{i} = 4. We will use Theorem B and the assumption about odd class numbers to prove that V_{1} ≅ V_{2}. This is the only case we need to discuss to prove Corollary 2.4. If we started instead with V_{1} ⊕ R_{−}^{2} ∼_{t} V_{2} ⊕ R_{−}^{2}, then stabilizing with R_{+} and applying Theorem 10.1 would reduce to the case above.

We may assume that q > 1 is a prime, and that the free representations have the form V_{1} = t + t^{i} and V_{2} = t^{1+2q} + t^{i+2q} where (i, 2q) = 1 (not all weights are ≡ 1 mod 4). The Reidemeister torsion invariant is Δ(V_{1})/Δ(V_{2}) = U_{1,i} ∈ Wh(ZG) in the notation for units of ZG used in [5, p. 732]. With respect to the involution ¯ : t ↦ −t^{−1}, this element defines a class u_{1,i} = {U_{1,i}} ∈ H^{1}(Wh(ZG^{−})). Since the map L_{1}^{p}(ZG^{−}) → L_{1}^{p}(C_{R_{−},G}(Z)) is injective, it is enough to show that the image of u_{1,i} is non–trivial in H^{1}(Δ) (see diagram [I] (6.7)).

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

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

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

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

Lemma 11.1. For (j, 2q) = 1 and j ≡ 1 mod 4, the image i_{∗}(u_{1,j}) = ⟨−1⟩ ∈ H^{1}(A^{×}).

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

Next we need a computation:

Lemma 11.2. 0 ≠ ⟨−1⟩ ∈ H^{1}(A^{×}).

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

[figure: H^{0}(E^{×}); H^{0}(E^{×}/A^{×}); δ; ≈; H^{1}(A^{×}); H^{0}(Ê^{×}); H^{0}(Ê^{×}/Â^{×}); H^{1}(Â^{×})]

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

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

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

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

12. The normal invariant

In this section we collect some results about normal cobordisms of lens spaces with cyclic fundamental group G. Recall that a necessary condition for the existence of a similarity V_{1} ⊕ W ∼_{t} V_{2} ⊕ W is that S(V_{1}) and S(V_{2}) must be s-normally cobordant. In [4, §1] it is asserted that their formula (A′) gives necessary and sufficient conditions for homotopy equivalent lens spaces to be s–normally cobordant, when G = C(2^{r}). However in [30, 1.3] it is shown that the given conditions (A′) are sufficient but not necessary. We only use the sufficiency here, and study the subgroups R̃^{free}(G) and R̃h^{free}(G) of R(G) defined in Section 3. Recall that R̃^{free}(G) = ker(Res: R^{free}(G) → R^{free}(G_{odd})).

Lemma 12.1. Let G = C(2^{r}q) be a finite cyclic group, and G_{2} be the 2–Sylow subgroup. If q > 1 there is an exact sequence

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

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

Proof. There is a short exact sequence

0 → R̃h^{free}(G) → R̃^{free}(G) → (Z/2^{r}q)^{×}/{±1}

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

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

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

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

Lemma 12.2. Let G = C(2^{r}q) be a finite cyclic group. Then

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

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

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

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

Example 12.3. For G = C(24) we have R̃^{free}(G) = {t−t^{5}, t−t^{7}, t−t^{11}} and the subgroup ker Res_{C(8)} ∩R̃^{free}(G) = {t − t^{7}, t^{5} − t^{11}}.

Lemma 12.4. Let G = C(2^{r}q) be a finite cyclic group, and G_{2} be the 2–Sylow subgroup. If r ≥ 2 there is an isomorphism

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

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

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

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

is surjective, and therefore

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

is also surjective.

Now suppose α ∈ R̃h^{free}(G) and Res_{G_{2}}(α) ∈ R̃n^{free}(G_{2}). This means that Res_{G_{2}}(α) = (W − W′) for some free G_{2}–representation W, W′ such that W ≏ W′. By the construction of the last paragraph, we can find free G–representations V, V′ such that (i) Res_{G_{2}}(V) = W and Res_{G_{2}}(V′) = W′, and (ii) α′ = (V − V′) ∈ R̃h^{free}(G). It follows that V ≏ V′ and so

α′ ∈ R̃n^{free}(G)), and

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

By Lemma 12.2, α − α′ ∈ R̃n^{free}(G) and so α ∈ R̃n^{free}(G). □

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

Lemma 12.5. Let b_{i,s} = (t^{i} − t^{2^{r−s}q−i}) ∈ R̃^{free}(G) for 1 ≤ i < 2^{r−s−1}q, 1 ≤ s ≤ r − 1, and (i, 2q) = 1. Let l(i, s) = s for 1 ≤ s ≤ r − 2, and l(i, r − 1) the order of i/(2q − i) ∈ (Z/2^{r})^{×}. Then b_{i,s} is an element of order 2^{l(i,s)} in R(G)/R_{h}(G), and 2^{l(i,s)} · b_{i,s} ≏ 0.

Proof. If 1 ≤ s ≤ r − 2 then k(b_{i,s}) = i/(2^{r−s}q − i) ∈ (Z/2^{r}q)^{×}/{±1} is congruent to −1 mod 4q. By Lemma 12.1 the linear span of these elements in R(G)/R_{h}(G) injects into R(G_{2})/R_{h}(G_{2}), where G_{2} is the 2–Sylow subgroup. By Lemma 12.4 the normal invariant is also detected by restriction to G_{2}, so it is enough to prove the assertions about these elements (s ≤ r − 2) when G = C(2^{r}).

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

ν_{2}(r!) = r − α_{2}(r),

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

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

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

Next we consider the normal invariant. For 1 ≤ s ≤ r − 2 we will take q = 1 and apply the criterion (A′) of [4] to show that 2^{s} · b_{i,s} ≈ 0. This amounts to a re-labelling of our original elements b_{i,s} without changing the order of their k–invariants.

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

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

so on the right-hand side we have

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

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

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

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

But by writing

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

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

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

If s = r − 1, then k(b_{i,s}) = i/(2q − i) ≡ −1 mod q, and k(b_{i,s}) ≡ 1 mod 2^{r} whenever i ≡ q mod 2^{r−1}. This is for example always the case for G = C(4q). Such elements b_{i,r−1} lie in the kernel of the restriction map to R(C(2^{r})). Moreover, if b_{i,r−1} ∈ ker(R(G) → R(C(2^{r}))), then 2b_{i,r−1} ∈ R_{h}(G) and 2b_{i,r−1} ≏ 0 by Lemma 12.2. Otherwise, the order of b_{i,r−1} ∈ R(G)/R_{h}(G) is the same as the order of its restriction Res(b_{i,r−1}) to the 2–Sylow subgroup, and Res(b_{i,r−1}) = ± Res(b_{j,s}) for l(i, r − 1) = s ≤ r − 2 and some j. □

Remark 12.6. As pointed out by the referee, Lemma 12.1 and Lemma 12.4 together give a short exact sequence

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

assuming that G is not a cyclic 2-group. It is not difficult to see from the proof of Theorem E (given in [I]), that for G_{2} = C(2^{r}), and r ≧ 4, the term R̃^{free}(G_{2})/R̃n^{free}(G_{2}) is the quotient of R̃Top^{free}(C(2^{r})) by the subgroup ⟨α_{1} + β_{1}⟩.

13. The proof of Theorem D

We first summarize our information about R_{Top}(G), obtained by putting together results from previous sections. If α ∈ R̃h,Top^{free}(G), we define the normal invariant order of α to be the minimal 2–power such that 2^{t}α ∈ R̃n,Top^{free}(G).

Theorem 13.1. Let G = C(2^{r}q), with q odd, and r ≧ 2.

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

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

is at most 2–primary. (iii) The subgroup R̃n,Top^{free}(G) has exponent two, and the Galois action induced by group

automorphisms is the identity. (iv) For any α ∈ R̃h,Top^{free}(G), if the normal invariant order of Res_{H}(α) is 2^{t}, then the

normal invariant order of α is 2^{t+1}.

Remark 13.2. In part (ii), φ(q) is the Euler function. The precise number of Z/2 summands in R̃n,Top^{free}(G) is determined by working out the conditions in Theorem C on the basis elements of R̃n^{free}(G). In cases where the conditions in Theorem C can actually be evaluated, the structure of R_{Top}(G) will thus be determined completely.

Proof. Parts (i) and (ii) of Theorem 13.1 have already been proved in Corollary 4.2, so it remains to discuss parts (iii) and (iv). In fact, the assertion that R̃n,Top^{free}(G) has exponent 2 is an immediate consequence of Theorem C. To see this, suppose that (V_{1} − V_{2}) is any element in R̃n^{free}(G). The obstruction to the existence of a stable non–linear similarity V_{1} ≈_{t} V_{2} is determined by the class {Δ(V_{1})/Δ(V_{2})} in the Tate cohomology group H^{1}(Wh(ZG^{−})/ Wh(ZH)), which has exponent 2. Since the Reidemeister torsion is multiplicative, Δ(V_{1} ⊕ V_{1}) = Δ(V_{1})^{2}, and we conclude that V_{1} ⊕ V_{1} ≈_{t} V_{2} ⊕ V_{2} by Theorem C. Finally, suppose that α ∈ R̃n,Top^{free}(G) and that σ is a group automorphism of G. By induction, we can assume that Res_{H}(α − σ(α)) = 0. We now apply Theorem C to β = α − σ(α), with W a complex G–representation such that W^{G} = 0, containing all the non-trivial irreducible representations of G with isotropy of 2-power index. Then β is detected by the image of its Reidemeister torsion invariant in H^{1}(Δ_{W ×R_{−}}). But by [I], Lemma 8.2, the Galois action on this group is trivial. Therefore β = α − σ(α) = 0.

For part (iv) we recall that the normal invariant for G lies in a direct sum of groups H^{4i}(G; Z_{(2)}) ≅ Z/2^{r}. Since the map Res_{H} induces the natural projection Z/2^{r} → Z/2^{r−1} on group cohomology, the result follows from [1, 2.6]. □

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

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

where W has no R_{+} or R_{−} summands, and l, s ≧ 1. But by [I], Corollary 6.13 we may assume that s = 1, and by Theorem 10.1 that l = 1, so we are reduced to the situation handled by Theorem C. The algebraic indeterminacy given there is computable, but not very easily if the associated cyclotomic fields have complicated ideal class groups. We carry out the computational details in one further case of interest (Theorem D).

The proof of Theorem D. We have a basis

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

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

Now suppose that V_{1} ⊕ W ⊕ R_{−} ⊕ R_{+} ∼_{t} V_{2} ⊕ W ⊕ R_{−} ⊕ R_{+}, for some complex G– representation W. Then Res_{H} V_{1} ≅ Res_{H} V_{2} since q is odd, and by Theorem C we get a similarity of the form V_{1} ⊕ R_{−}^{l} ⊕ R_{+} ∼_{t} V_{2} ⊕ R_{−}^{l} ⊕ R_{+}. But by Theorem 10.1 this implies that V_{1} ⊕ R_{−} ⊕ R_{+} ∼_{t} V_{1} ⊕ R_{−} ⊕ R_{+}. Therefore, for any element a_{i} = (t^{i} − t^{i+2q}) ∈ B we have 2a_{i} ∉ Rt^{free}(G) but 4a_{i} ∈ Rt^{free}(G). However, in Section 11 we determined the bounded surgery obstructions for all these elements. Since i_{∗}(u_{1,j}) = ⟨−1⟩ ∈ H^{1}(A^{×}) for all j with (j, 2q) = 1 by Lemma 11.1, there are further stable relations a_{1} + a_{j} ≈ a_{1} + a_{k}, or a_{1} ≈ a_{j} for all j. It follows that a basis for R̃Top^{free}(G) is given by {a_{1} | 4a_{1} ≈ 0}. □

References

[1] S. E. Cappell and J. L. Shaneson, Non-linear similarity, Ann. of Math. (2) 113 (1981), 315–355.

[2] , The topological rationality of linear representations, Inst. Hautes Études Sci. Publ. Math. 56 (1983), 309–336.

[3] , Torsion in L-groups, Algebraic and Geometric Topology, Rutgers 1983, Lecture Notes in Mathematics, vol. 1126, Springer, 1985, pp. 22–50.

[4] S. E. Cappell, J. L. Shaneson, M. Steinberger, S. Weinberger, and J. West, The classification of nonlinear similarities over ℤ/2^{r}, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 51–57.

[5] S. E. Cappell, J. L. Shaneson, M. Steinberger, and J. West, Non-linear similarity begins in dimension six, J. Amer. Math. Soc. 111 (1989), 717–752.

[6] M. Cárdenas and E. K. Pedersen, On the Karoubi filtration of a category, K-theory 12 (1997), 165–191.

[7] G. Carlsson and E. K. Pedersen, Controlled algebra and the Novikov conjectures for K- and L-theory, Topology 34 (1995), 731–758.

[8] D. Carter, Localizations in lower algebraic K-theory, Comm. Algebra 8 (1980), 603–622.

[9] A. W. M. Dress, Induction and structure theorems for orthogonal representations of finite groups, Ann. of Math. (2) 102 (1975), 291–325.

[10] I. Hambleton and E. K. Pedersen, Bounded surgery and dihedral group actions on spheres, J. Amer. Math. Soc. 4 (1991), 105–126.

[11] , Non-linear similarity revisited, Prospects in Topology: (Princeton, NJ, 1994), Annals of Mathematics Studies, vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 157–174.

[12] I. Hambleton, L. Taylor, and B. Williams, An introduction to the maps between surgery obstruction groups, Algebraic Topology (Aarhus, 1982), Lecture Notes in Mathematics, vol. 1051, Springer, Berlin, 1984, pp. 49–127.

[13] I. Hambleton and L. R. Taylor, A guide to the calculation of surgery obstruction groups, Surveys in Surgery Theory, Volume I, Annals of Mathematics Studies, vol. 145, Princeton Univ. Press, 2000, pp. 225–274.

[14] W-C. Hsiang and W. Pardon, When are topologically equivalent representations linearly equivalent, Invent. Math. 68 (1982), 275–316.

[15] I. Madsen and M. Rothenberg, On the classification of G-spheres I: equivariant transversality, Acta Math. 160 (1988), 65–104.

[16] R. J. Milgram and A. A. Ranicki, The L-theory of Laurent extensions and genus 0 function fields, J. Reine Angew. Math. 406 (1990), 121–166.

[17] R. Oliver, Whitehead Groups of Finite Groups, London Math. Soc. Lecture Notes, vol. 132, Cambridge Univ. Press, 1988.

[18] E. K. Pedersen and C. Weibel, A nonconnective delooping of algebraic K-theory, Algebraic and Geometric Topology, (Rutgers, 1983), Lecture Notes in Mathematics, vol. 1126, Springer, Berlin, 1985, pp. 166–181.

[19] A. A. Ranicki, The double coboundary, Letter to B. Williams, available at the web page http://www.maths.ed.ac.uk/ aar/surgery/brucelet.pdf, 1981.

[20] , Exact Sequences in the Algebraic Theory of Surgery, Math. Notes, vol. 26, Princeton Univ. Press, 1981.

[21] , Additive L-theory, K-theory 3 (1989), 163–195.

[22] , Algebraic L-theory and Topological Manifolds, Cambridge Tracts in Math., vol. 102, Cambridge Univ. Press, 1992.

[23] G. de Rham, Sur les nouveaux invariants topologiques de M Reidemeister, Mat. Sbornik 43 (1936), 737–743, Proc. International Conference of Topology (Moscow, 1935).

[24] , Reidemeister’s torsion invariant and rotations of S^{n}, Differential Analysis, (Bombay Colloq.), Oxford University Press, London, 1964, pp. 27–36.

[25] J.-P. Serre, Linear Representations of finite groups, Graduate Texts in Mathematics, vol. 42, Springer, 1977.

[26] R. Swan, Induced representations and projective modules, Ann. of Math. (2) 71 (1960), 267–291.

[27] R. G. Swan and E. G. Evans, K-theory of finite groups and orders, Lecture Notes in Mathematics, vol. 149, Springer, 1970.

[28] C. T. C. Wall, Surgery on Compact Manifolds, Academic Press, New York, 1970.

[29] , On the classification of Hermitian forms. II. Semisimple rings, Invent. Math. 18 (1972), 119– 141.

[30] C. Young, Normal invariants of lens spaces, Canad. Math. Bull. 41 (1998), 374–384.

The first part of this paper is:

I. Hambleton and E. K. Pedersen, Topological equivalence of linear representations for cyclic groups: I, Ann. of Math. (2) (to appear).

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

E-mail address: ian@math.mcmaster.ca

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

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