”Surgery Obstructions of Fibre Bundles” by Wolfgang Lück and Andrew Ranicki

Introduction

Chern, Hirzebruch and Serre [6] proved that the signature of the total space of a fibre bundle F ⟶ E ⟶^{p} B in which the fundamental group π_{1}(B) acts trivially on H^{∗}(F; R) is the product of the signatures of the fibre and base

sign(E) = sign(F) sign(B) ∈ Z ,

with sign = 0 for manifolds of dimension ≢ 0(mod 4). Kodaira [13], Atiyah [3] and Hirzebruch [12] constructed various examples of fibre bundles in which π_{1}(B) acts non-trivially on H^{∗}(F; R) and the signature is not multiplicative. Moreover, in the case where both B and F are even-dimensional the Hirzebruch signature theorem

sign(E) = ⟨L(E), [E]⟩ ∈ Z

and the Atiyah-Singer index theorem were used by Atiyah [3] to obtain a characteristic class formula for the signature of E involving a contribution from the action of π_{1}(B) on H^{∗}(F; R). The flat (−1)^{k}-symmetric bundle Γ over B with fibres H^{k}(F_{x}; R) (x ∈ B, k = dim(F)/2), has a real (resp. complex) K-theory signature [Γ]_{K} ∈ KO(B) for k ≡ 0(mod 2) (resp. KU(B) for k ≡ 1(mod 2)) and the twisted signature theorem is

sign(E) = ⟨ch([Γ]_{K}) ∪ L̃(B), [B]⟩ ∈ Z .

Lusztig [19] and Meyer [20] extended this expression to the Γ-twisted signature sign(B, Γ) ∈ Z for any sheaf Γ of (−1)^{k}-symmetric forms over an even-dimensional manifold B.

In this paper we apply the algebraic surgery transfer of Lück and Ranicki [18] for a fibration F ⟶ E ⟶^{p} B with the fibre F a d-dimensional Poincaré complex

p^{∗} : L_{n}(Zπ_{1}(B)) ⟶ L_{n+d}(Zπ_{1}(E))

to a further investigation of the behaviour of the Wall surgery obstruction and the Mishchenko symmetric signature in fibrations. These invariants are generalizations of the equivariant signature, and the characteristic classes have to be replaced by more general L-theory invariants. Instead of dealing with the action of π_{1}(B) on H^{∗}(F; R) we consider the chain homotopy action of π_{1}(B) on the chain complex C(F) induced by the homotopy action of π_{1}(B) on F given by the fibre transport ω : π_{1}(B) ⟶ [F, F].

Generalizing the work of Dress [8] and Yoshida [39] for finite groups we study the equivariant L-groups L^{d}(π, Z) of a group π. For d = 2k (resp. 2k+1) L^{d}(π, Z) is the Witt group of nonsingular π-equivariant (−1)^{k}-symmetric forms (resp. linking forms) on finitely generated free (resp. finite) abelian groups. The equivariant Witt groups L^{2k}(π, Z) are well-known, and their applications to group actions, knot theory and the surgery obstruction groups of finite groups have motivated extensive computations, both for finite groups (Alexander, Conner and Hamrick [1]) and infinite groups (Neumann [25]). In the main body of the text we shall take full account of the various orientations: in the introduction we suppose for simplicity that all manifolds and Poincaré complexes are orientable.

The tensor product over Z with the diagonal π-action induces a pairing

⊗ : L^{d}(π, Z) ⊗ L_{n}(Zπ) ⟶ L_{n+d}(Zπ) .

A fibration F ⟶ E ⟶^{p} B with d-dimensional Poincaré fibre F has a π_{1}(B)-equivariant symmetric signature invariant

σ^{∗}(F, ω) ∈ L^{d}(π_{1}(B), Z) ,

given for d = 2k (resp. 2k + 1) by the Witt class of the nonsingular π_{1}(B)-equivariant intersection form on H^{k}(F) (resp. linking form on the torsion subgroup of H^{k+1}(F)). It depends only on the fibre F and the fibre transport ω : π_{1}(B) ⟶ [F, F]. Our main result expresses the composition

p_{∗} ∘ p^{∗} : L_{n}(Zπ_{1}(B)) ⟶^{p^{∗}} L_{n+d}(Zπ_{1}(E)) ⟶^{p_{∗}} L_{n+d}(Zπ_{1}(B))

as product with σ^{∗}(F, ω).

Theorem 2.7 (up-down formula) For a fibration F ⟶ E ⟶^{p} B with F a d-dimensional geometric Poincaré complex we have:

p_{∗} ∘ p^{∗} = σ^{∗}(F, ω) ⊗ − : L_{n}(Zπ_{1}(B)) ⟶ L_{n+d}(Zπ_{1}(B)) .

In the appendix to this paper the algebraic L-theory assembly map of Ranicki [32] is used to relate this expression for p_{∗} ∘ p^{∗} to the characteristic class formula for the signature of the total space of a fibre bundle.

Although we shall be mainly concerned with fibrations of connected spaces, it should be noted that the up-down formula 2.7 also applies to finite covers, with the fibre F a genuinely finite 0-dimensional Poincaré complex and π_{1}(E) → π_{1}(B) the inclusion of a subgroup of finite index. As in the general case, the element σ^{∗}(F, ω) ∈ L^{0}(π_{1}(B), Z) is represented by the π_{1}(B)-action on the nonsingular symmetric form (H^{0}(F), ⊕1) of the fibre. In fact, the up-down formula in this case has already been obtained in Hambleton, Ranicki and Taylor [11].

The up-down formula simplifies considerably for fibrations which are orientable, i.e. with trivial fibre transport ω : π_{1}(B) ⟶ [F, F] , generalizing the multiplicativity of the signature in this case. Namely, the {1}-equivariant Witt group L^{d}({1}, Z) is just the simplyconnected symmetric L-group L^{d}(Z) of Ranicki [29]

L^{d}(\{1\}, Z) = L^{d}(Z) = \{ \begin{matrix}Z (signature) & if d ≡ 0(mod 4) \\ Z_{2} (deRham invariant) & if d ≡ 1(mod 4) \\ 0 & otherwise . \end{matrix}

For an orientable fibration σ^{∗}(F, ω) ∈ L^{d}(π_{1}(B), Z) is the image of the simply-connected symmetric signature σ^{∗}(F) ∈ L^{d}(Z). Combining 2.7 with the factorization

⊗ : L^{d}(Z) ⊗ L_{n}(Zπ) ⟶ L^{d}(π, Z) ⊗ L_{n}(Zπ) ⟶ L_{n+d}(Zπ)

gives:

Corollary 2.10 For an orientable fibration F ⟶ E ⟶^{p} B with F a d-dimensional geometric Poincaré complex

p_{∗} ∘ p^{∗} = σ^{∗}(F) ⊗ − : L_{n}(Zπ_{1}(B)) ⟶ L_{n+d}(Zπ_{1}(B))

is multiplication by the simply-connected symmetric signature σ^{∗}(F) ∈ L^{d}(Z).

The significance of the algebraic surgery transfer is due to the fact that it does describe the geometric surgery transfer which is defined as follows. Let F ⟶ E ⟶^{p} B be a smooth fibre bundle of connected smooth compact manifolds such that F is closed. Let d and n be the dimensions of F and B. An element λ in L_{n}(Zπ_{1}(B)) is represented by a normal map f : M ⟶ N together with a reference map g : N ⟶ B such that f induces a homotopy equivalence on the boundaries. (We suppress bundle data). The pull back construction yields a surgery problem \overline{f} : \overline{M} ⟶ \overline{N} with a reference map \overline{g} : \overline{N} ⟶ E. Its class in L_{d+n}(Zπ_{1}(E)) is defined to be the image of λ under the geometric surgery transfer of Quinn [27]

p^{!} : L_{n}(Zπ_{1}(B)) ⟶ L_{n+d}(Zπ_{1}(E))

The significance of the geometric surgery transfer is obvious from its definition. It gives the possibility of proving the vanishing of surgery obstructions. The strategy consists of two steps. Firstly, show that the target of a given surgery problem is the total space of an appropriate fibre bundle such that the given surgery problem is the pull back of a surgery problem for the base space. Secondly, show that the surgery transfer vanishes. Hence vanishing theorems are of particular importance. We derive from the up-down formula:

Corollary 6.1 Let G be a compact connected d-dimensional Lie group which is not a torus. Let G ⟶ E ⟶^{p} B be a G-principal bundle. Then

p^{∗} = p^{!} : L_{n}(Zπ_{1}(B)) ⟶ L_{n+d}(Zπ_{1}(E))

is trivial.

Surgery transfers appear naturally in the study of group actions, namely, as surgery transfers of the normal sphere bundles of the fixed point sets. In Lück-Madsen [16] and [17] a spectral sequence is constructed which converges to the equivariant surgery obstruction group and whose E^{2}-term consists of algebraic L-groups. Its differentials are given by surgery transfers of the normal sphere bundles of the fixed point sets. For transformation groups of odd order the spectral sequence collapses. This is not true in the even order case, as already shown by explicit computations for Z/2. Similar spectral sequences occur in the isovariant transverse linear setting of Browder and Quinn [5]. Using these spectral sequences and periodicity results Dovermann and Schultz compare the equivariant and isovariant transverse linear setting in [7].

We shall also give some computations for rational coefficients Q instead of integer coefficients Z in section 4. The corresponding L-theory change of ring homomorphism is an isomorphism if one inverts 2, so that rational computations still give good information about integral ones.

A fibration is called untwisted if the pointed fibre transport ω^{+} : π_{1}(E) ⟶ [F, F]^{+} is trivial. We give further vanishing results for this class of fibrations in section 5. This includes orientable fibrations with H-spaces as fibre. The class of untwisted fibrations seems to be the largest class of fibrations where little information about the fibre suffices to prove vanishing results for the surgery transfer.

We are mainly dealing with L-groups of finitely generated free modules, or in other words, with L^{h}. In section 7 we briefly state the necessary modifications which have to be made when dealing with “decorated” L-groups like L^{p} or L^{s}.

We have tried to postpone the main technical parts to the last four sections. In particular it is not necessary to know the construction of the algebraic surgery transfer before section 8, as it only appears in the rather technical proof of the up-down formula 2.7 in sections 9 and 10. We recall which algebraic data are needed to define the algebraic surgery transfer and how the pointed fibre transport provides these data in section 1. Commencing with section 8 we assume that the reader is somewhat familiar with the approach to L-theory using Poincaré chain complexes as developed in Ranicki [30].

We make some comments on the proof of the up-down formula. The proof presented here is fairly straightforward. This has the advantage that one does avoid some machinery and has not to introduce new notions and the disadvantage that the conceptual ideas are somewhat hidden. The motivation for the proof in the L-theory case comes from the proof in the K-theory case as developed in Lück [15]. The relevant products fit together in the following commutative diagram for i = 0, 1:

K_{0}^{c}(Zπ_{1}(B) − Zπ_{1}(E)) ⊗ K_{i}(Zπ_{1}(B)) \underrightarrow{⊗}  K_{i}(Zπ_{1}(E))
i ⊗ id ↑  ↑ id
K_{0}^{c}(Δ − π_{1}(E), Z) ⊗ K_{i}(Zπ_{1}(B))  \underrightarrow{⊗_{t}}  K_{i}(Zπ_{1}(E))
p_{∗} ⊗ id ↓  ↓ p_{∗}
K_{0}^{c}(\{1\} − π_{1}(B), Z) ⊗ K_{i}(Zπ_{1}(B))  \underrightarrow{⊗_{t}}  K_{i}(Zπ_{1}(B))
e ⊗ id ↓  ↓ id
Sw(π_{1}(B), Z) ⊗ K_{i}(Zπ_{1}(B))  \underrightarrow{⊗_{Z}}  K_{i}(Zπ_{1}(B))

such that up to isomorphism

K_{0}^{c}(Zπ_{1}(B) − Zπ_{1}(B)) = K_{0}^{c}({1} − π_{1}(B), Z) = Sw(π_{1}(B), Z) .

The group K_{0}^{c}(Zπ_{1}(B) − Zπ_{1}(E)) is the Grothendieck group of chain homotopy representations of Zπ_{1}(B) in Zπ_{1}(E) and K_{0}^{c}(Δ−π_{1}(E), Z) is the Grothendieck group of ZΔ-chain complexes with a π_{1}(E)-twist, where Δ is the kernel of π_{1}(p) : π_{1}(E) ⟶ π_{1}(B). Sw(π_{1}(B), Z) is the Grothendieck-Swan group of Zπ_{1}(B)-modules which are finitely generated and free as abelian groups. The pairing ⊗_{Z} is given by the tensor product over Z together with the diagonal action. Now the fibre together with the fibre transport define an element [F, ω]_{1} in K_{0}^{c}(Zπ_{1}(B) − Zπ_{1}(E)), and the K-theory transfer p^{∗} is given by the evaluation of the top pairing on [F, ω]_{1}. It turns out that [F, ω]_{1} is the image of an element [F, ω]_{2} ∈ K_{0}^{c}(Δ − π_{1}(E), Z) under the canonical map i. Let [F, ω] ∈ K_{0}^{c}({1} − π_{1}(B), Z) be the image of [F, ω]_{2} under the map p_{∗} induced by π_{1}(p). As the diagram above commutes p_{∗} ∘ p^{∗} agrees with [F, ω]⊗_{t}?. An element in K_{0}^{c}({1} − π_{1}(B), Z) is given by a chain complex C of finitely generated free Z-modules, with a homotopy π_{1}(B)-action. The homology groups H_{i}(C) are Zπ_{1}(B)-modules which are finitely generated as abelian groups. Using finitely generated free Z-resolutions one obtains elements [H_{i}(C)] in Sw(π_{1}(B), Z). The isomorphism e : K_{0}^{c}({1} − π_{1}(B), Z) ⟶ Sw(π_{1}(B), Z) sends the class of C to the K-theory Euler characteristic ∑_{i}(−1)^{i} · [H_{i}(C)]. Hence p_{∗} ∘p^{∗} is given by the evaluation of the bottom pairing ⊗_{Z} on the element ∑_{i}(−1)^{i} · [H_{i}(F)] in Sw(π_{1}(B), Z). The up-down formula 2.7 is just the L-theory version of this K-theory formula. The equivariant Witt group L^{d}(π_{1}(B), Z) plays the role of Sw(π_{1}(B), Z), and the π_{1}(B)-equivariant signature σ^{∗}(F, ω) corresponds to ∑_{i}(−1)^{i} · [H_{i}(F)].

We shall construct the L-theory analogue of this diagram in a subsequent paper, obtaining a Witt group for the transfer p^{∗} rather than just p_{∗} ∘p^{∗}. This is not a trivial problem, since it involves higher coherences of the homotopy action of π_{1}(B) on the fibre. One does not see this difficulty in homology, since any homotopy action on a space induces an honest action in homology. By sticking to the middle dimensions our proof of the up-down formula is essentially on the homology level, and so avoids higher coherences.

After this paper was written we received the preprint Yan [38], which obtains the up-down formula 2.7 in the special case π_{1}(F) = {1}, d = 2k.

The paper is organized as follows :

0. Introduction
1. Algebraic Surgery Transfer
2. The up-down Formula for p_{∗} ∘ p^{∗}
3. Symmetric Signature
4. Rational Computations
5. Untwisted Fibrations
6. G-Principal Bundles
7. Change of K-theory
8. Review of the Construction of Algebraic Surgery Transfer
9. Proof of the up-down Formula in Even Base Dimensions
10. Proof of the up-down Formula in Odd Base Dimensions
11. The Pairing 2.4 is Well-Defined

Appendix: Characteristic Class Formulae References

1. Algebraic Surgery Transfer

We explain which algebraic data are needed to define the algebraic surgery transfer and how these data can be obtained from the pointed fibre transport.

A ring with involution is an associative ring R with unit 1 together with a function ¯ : R ⟶ R satisfying \overline{\overline{a}} = a, \overline{a + b} = \overline{a} + \overline{b}, \overline{ab} = \overline{b}\overline{a} and \overline{1} = 1 for a, b ∈ R. Given a group π together with a homomorphism w : π ⟶ {±1}, the w-twisted involution on Zπ is given by \overline{g} = w(g) · g^{−1}. Let R and S be rings with involution. Let C be a d-dimensional finitely generated free S-chain complex. Let C^{d−∗} be the S-chain complex hom_{S}(C_{d−∗}, S), where the involution on S is used to define a left S-structure on hom_{S}(C_{d−∗}, S). We shall identify (C^{d−∗})^{d−∗} and C. Consider an S-chain equivalence α : C^{d−∗} ⟶ C such that α^{d−∗} and α are chain homotopy equivalent. Then the set [C, C]_{S} of chain homotopy classes of self chain maps of C becomes a ring with involution [f] ↦ [α ∘ f^{d−∗}α^{−1}]. Denote by [C, C]_{S}^{op} the opposite ring. Let D_{d}(S) be the additive category with involution having d-dimensional finitely generated free S-chain complexes as objects and homotopy classes of chain maps as morphisms. If U : R ⟶ [C, C]_{S}^{op} is a homomorphism of rings with involution, we call (C, α, U) a symmetric representation of R into D_{d}(S). The surgery transfer

1.1 (C, α, U)^{∗} : L_{n}(R) ⟶ L_{n+d}(S)

associated to (C, α, U) is defined in Lück-Ranicki [18] and will be reviewed in section 8.

A d-dimensional finite Poincaré complex F = (F, [F], w(F)) consists of a finite CW complex F, an orientation homomorphism w(F) : π_{1}(F) ⟶ {±1} and a fundamental class [F] ∈ H^{d}(F,^{w(F)} Z) such that ∩[F] : C^{d−∗}(F̃) ⟶ C_{∗}(F̃) is a Zπ_{1}(F)-chain homotopy equivalence. Here we use the w(F)-twisted involution to define a left Zπ_{1}(F)-structure on C^{d−∗}(F̃) and ^{w(F)}Z is the Zπ_{1}(F)-module given by Z and the homomorphism w(F). Any closed manifold is a Poincaré complex. Let F ⟶ E ⟶^{p} B be a fibration of (well-pointed) connected spaces such that F is a d-dimensional Poincaré complex. We do not assume that B and E are manifolds. Let F̂ ⟶ F be the pull back of the universal covering of E with the inclusion of the fibre. Denote by [F, F] (resp. [F, F]^{+}) the monoid of (pointed) homotopy classes of (pointed) self-maps of F. Let [F̂, F̂]_{π_{1}(E)} be the monoid of π_{1}(E)-homotopy classes of π_{1}(E)-self-maps of F̂. The elementary construction of the monoid homomorphisms

1.2

τ : π_{1}(F) ⟶ [F, F]^{+}  (operation of the fundamental group)
ω^{+} : π_{1}(E) ⟶ [F, F]^{+}  (pointed fibre transport)
ω : π_{1}(B) ⟶ [F, F]  (fibre transport)
ω̂ : π_{1}(B) ⟶ [F̂, F̂]_{π_{1}(E)}  (equivariant fibre transport)

based on the homotopy lifting property can be found in Lück [14], section 6.

Let H^{d}(F̃) be H_{0}(hom_{Zπ_{1}(F)}(C_{d−∗}(F̃), Zπ_{1}(F))) where hom_{Zπ_{1}(F)}(C_{d−∗}(F̃), Zπ_{1}(F)) has the untwisted Zπ_{1}(F)-structure (wϕ)(x) = ϕ(x)w^{−1}. Consider a self-map f : F ⟶ F. Choose a lift f̃: F̃ ⟶ F̃. Let H^{d}(f̃) : H^{d}(F̃) ⟶ H^{d}(F̃) be given by the Zπ_{1}(f)^{−1}-equivariant chain map hom_{Zπ_{1}(F)}(C(f̃), Zπ_{1}(f)^{−1}). By Poincaré duality H^{d}(F̃) is isomorphic to ^{w(F)}Z. Hence w(F) depends only on the homotopy type of F and we have w(F) ∘ π_{1}(f) = w(F). The homotopy orientation homomorphism v(F) : [F, F]^{+} ⟶ {±1} of F sends [f] to the degree of H^{d}(f̃). Define

1.3 \begin{aligned}v̂ = v(F) ∘ ω^{+} & : & π_{1}(E) & ⟶ & \{±1\} \\ ŵ = w ∘ π_{1}(p) & : & π_{1}(E) & ⟶ & \{±1\} \end{aligned}

Lemma 1.4 1. The following diagram commutes

π_{1}(F) ⟶^{π_{1}(i)} π_{1}(E) ⟶^{π_{1}(p)} π_{1}(B) ⟶ {1}
↓ id  ↓ ω^{+}  ↓ ω
π_{1}(F) ⟶^{τ} [F, F]^{+} \underrightarrow{pr} [F, F] ⟶ \{1\}

The upper row is an exact sequence of groups. The lower row is exact in the sense that pr is surjective and pr(f_{0}) = pr(f_{1}) holds for f_{0}, f_{1} ∈ [F, F]^{+}, if and only there is g ∈ π_{1}(F) satisfying f_{0} = τ(g)f_{1}.

2. The π_{1}(E)-space F̂ can be identified with π_{1}(E)×_{π_{1}(F)} F̃. Let g ∈ π_{1}(B) and ĝ ∈ π_{1}(E) be any lift of g. Then ω̂(g) is given by r(ĝ) ×_{π_{1}(F)} ω^{+}(ĝ^{−1}) where r(ĝ) : π_{1}(E) → π_{1}(E) is right multiplication with ĝ.

3. The composition of v̂ with π_{1}(i) : π_{1}(F) ⟶ π_{1}(E) is the first Stiefel-Whitney class w(F). If F ⟶ E ⟶^{p} B is a smooth fibre bundle of compact manifolds and w is w(B), then ŵv̂ is w(E).

We get from ω̂ a ring homomorphism U : Zπ ⟶ [C_{∗}(F̂), C_{∗}(F̂)]_{Zπ_{1}(E)}^{op}. The Poincaré duality chain equivalence for C_{∗}(F̃) induces a Zπ_{1}(E)-chain equivalence α : C^{d−∗}(F̂) ⟶ C_{∗}(F̂). If we equip Zπ_{1}(B) resp. Zπ_{1}(E) with the w-twisted resp. v̂ŵ-twisted involution, (C(F̂), α, U) is a symmetric representation of Zπ_{1}(B) in D_{d}(Zπ_{1}(E)). The transfer (C(F̂), α, U)^{∗} introduced in 1.1 is the algebraic surgery transfer of the fibration

1.5 p^{∗} : L_{n}(Zπ_{1}(B), w(B)) ⟶ L_{n+d}(Zπ_{1}(E), w(E))

2. The up-down Formula for p_{∗} ∘ p^{∗}

In this section we want to analyse the composition p_{∗} ∘ p^{∗} in L-theory. Beforehand we review the computation of p_{∗} ∘ p^{∗} for the K-theory transfer p^{∗} : K_{n}(Zπ_{1}(B)) → K_{n}(Zπ_{1}(E)) for n = 0, 1 of a fibration F ⟶ E ⟶^{p} B of connected spaces with finitely dominated fibre F in order to motivate the following constructions. Recall that F is finitely dominated, if F is up to homotopy the retract of a finite CW complex. For π = π_{1}(B) let Sw(π, Z) be the Grothendieck group of Zπ-modules which are finitely generated and free as Z-modules. This is the abelian group having isomorphism classes of such modules as generators and any short exact (not necessarily split exact) sequence of such modules 0 → L → M → N → 0 gives a relation [L] − [M] + [N] = 0. The tensor product over Z with the diagonal π-action induces a pairing

2.1 ⊗ : Sw(π, Z) ⊗ K_{n}(Zπ) ⟶ K_{n}(Zπ)

This pairing was defined and analysed in Swan [34]. Consider a Zπ-module M which is finitely generated as Z-module. Let {0} ⟶ G_{1} ⟶ G_{0} ⟶ M ⟶ {0} be a 1-dimensional Zπ-resolution such that G_{0} and G_{1} are finitely generated and free as Z-module. Define a class [M] ∈ Sw(π, Z) depending only on the isomorphism type of M by [G_{0}] − [G_{1}]. The homotopy action of π on the fibre F given by the fibre transport ω : π_{1}(B) ⟶ [F, F] defines a Zπ-structure on H_{i}(F, Z). We obtain a class

2.2 [F] = ∑_{i≥0}(−1)^{i} · [H_{i}(F, Z)] ∈ Sw(π, Z)

Then p_{∗} ∘ p^{∗} coincides with [F]⊗? (see Lück [15], Munkholm [21], Munkholm-Pedersen [22]). The construction in L-theory is similar, but – as is nearly always the case – harder and the odd-dimensional case is more difficult than the even-dimensional.

We first explain the algebraic setup in even dimensions. Let π be a group and v, w : π ⟶ {±1} be group homomorphisms. For an integer k a nonsingular (−1)^{k}- symmetric form over Z is a finitely generated free Z-module M together with an isomorphism ψ : M ⟶ hom_{Z}(M, Z) such that (−1)^{k} · hom_{Z}(ψ, id) : M ⟶ hom_{Z}(M, Z) agrees with ψ. Here and elsewhere we identify M with hom_{Z}(hom_{Z}(M, Z), Z) using the natural isomorphism sending m to f ↦ f(m). Note that ψ is the same as a nonsingular (−1)^{k}-symmetric bilinear pairing \overline{ψ} : M ⊗ M ⟶ Z. If M additionally carries a left Zπ-structure such that \overline{ψ}(gx, gy) = v(g) · \overline{ψ}(x, y) holds for all x, y ∈ M and g ∈ π, we call (M, ψ) a nonsingular (π, v)-equivariant (−1)^{k}-symmetric form over Z. The last condition about \overline{ψ} is equivalent to the assumption that ψ : M ⟶ hom_{Z}(M, Z) is a Zπ-map, if π acts on hom_{Z}(M, Z) from the left by (gf)(x) = v(g) · f(g^{−1}x) for f : M ⟶ Z, g ∈ π and x ∈ M.

Let (M, ψ) be a nonsingular (π, v)-equivariant (−1)^{k}-symmetric form over Z. It is hyperbolic if there is a Zπ-module L ⊂ M such that M ⟶ hom_{Z}(L, Z) sending m to l ↦ ψ(m)(l) is an epimorphism with L as kernel. Note that L is finitely generated and free as a Z-module. We call two (π, v)-equivariant (−1)^{k}-symmetric forms over Z stably isomorphic if they become isomorphic after adding hyperbolic ones. Let L^{2k}(π, Z, v) be the Witt group of stable isomorphism classes of (π, v)-equivariant (−1)^{k}-symmetric forms over Z. Addition is given by direct sums and the inverse of (M, ψ) is represented by (M, −ψ). The tensor product over Z together with the diagonal π-action defines a pairing

2.3 ⊗ : L^{2k}(π, Z, v) ⊗ L_{n}(Zπ, w) ⟶ L_{n+2k}(Zπ, vw)

This is essentially the pairing of Fröhlich and McEvett [9], Thomas [35] and Dress [8]. It plays as important a role in induction theory for L-theory as Swan’s pairing 2.1 does for K-theory.

Next we deal with the odd-dimensional case. A nonsingular (−1)^{k+1}-symmetric linking form over Z consists of a finitely generated torsion Z-module M together with an isomorphism ψ : M ⟶ hom_{Z}(M, Q/Z) such that hom(ψ, id) : M ⟶ hom_{Z}(M, Q/Z) is (−1)^{k+1} · ψ, where we identify M with hom_{Z}(hom_{Z}(M, Q/Z), Q/Z). One may interpret ψ as a nonsingular (−1)^{k+1}-symmetric bilinear pairing \overline{ψ} : M ⊗ M ⟶ Q/Z. If M additionally carries a left Zπ-structure such that \overline{ψ}(gx, gy) = v(g) · \overline{ψ}(x, y) holds for all x, y ∈ M and g ∈ π, we call (M, ψ) a nonsingular (π, v)-equivariant (−1)^{k+1}-symmetric linking form over Z. The last condition about \overline{ψ} is equivalent to the assumption that ψ : M ⟶ hom_{Z}(M, Q/Z) is a Zπ-map, if π acts on hom_{Z}(M, Q/Z) from the left by (gf)(x) = v(g) · f(g^{−1}x) for g ∈ π and x ∈ M. A nonsingular (π, v)-equivariant (−1)^{k+1}-symmetric linking form over Z is hyperbolic if there is a Zπ-submodule L ⊂ M such that M ⟶ hom_{Z}(L, Q/Z) sending m to l ↦ ψ(m)(l) is an epimorphism with L as kernel. Note that L is a finitely generated torsion Z-module. Let (M, ψ) be a Q-nonsingular (π, v)-equivariant (−1)^{k+1}-symmetric form over Z, where Q-nonsingular means that ψ : M ⟶ hom_{Z}(M, Z) is rationally an isomorphism or, equivalently, has finitely generated torsion Z-modules as kernel and cokernel. Its boundary (∂M, ∂ψ) is the nonsingular (π, v)-equivariant (−1)^{k+1}-symmetric linking form over Z given by ∂M = cok(ψ) and

∂ψ : cok(ψ) ⟶ hom_{Z}(cok(ψ), Q/Z) x ↦ (y ↦ \frac{x(z)}{s})

where x, y ∈ hom_{Z}(M, Z), z ∈ M and s ∈ Z\{0} satisfying sy = ψ(z). Two (π, v)- equivariant (−1)^{k+1}-symmetric linking forms over Z are called equivalent if they become isomorphic after adding hyperbolic ones and boundaries. Let L^{2k+1}(π, Z, v) be the Witt group of equivalence classes of nonsingular (π, v)-equivariant (−1)^{k+1}-symmetric linking forms over Z. Addition is given by direct sums and the inverse of (M, ψ) is represented by (M, −ψ). Next we define a pairing

2.4 ⊗ : L^{2k+1}(π, Z, v) ⊗ L_{n}(Zπ, w) ⟶ L_{n+2k+1}(Zπ, vw)

Let (M, ψ) be a nonsingular (π, v)-equivariant (−1)^{k+1}-symmetric linking form over Z. Choose a finitely generated free Z-resolution {0} ⟶ F_{1} ⟶^{f_{1}} F_{0} ⟶^{ϵ} M ⟶ {0}. We use the following isomorphism as an identification

ϵ̂: H_{0}(F^{1−∗}, Z) ⟶ hom_{Z}(M, Q/Z) \overline{g} ↦ (m ↦ \frac{g(sx)}{s})

where g ∈ hom_{Z}(F_{1}, Z), x ∈ F_{0}, s ∈ Z satisfying ϵ(x) = m and sx ∈ F_{1}. As hom_{Z}(F_{1−∗}, Z) and F are free Z-resolutions, there is a Z-chain equivalence α : hom_{Z}(F_{1−∗}, Z) ⟶ F_{∗} satisfying H(α) = ψ and α ≃ (−1)^{k} · hom_{Z}(ϕ, id). Let id ⊗ α : Zπ ⊗_{Z} hom_{Z}(F_{1−∗}, Z) ⟶ Zπ ⊗_{Z} F_{∗} be the induced Zπ-chain equivalence. The Zπ-structure on M yields a homomorphism V : π ⟶ [F_{∗}, F_{∗}]_{Z}^{op}. This gives a homomorphism U : Zπ → [Zπ ⊗_{Z} F_{∗}, Zπ ⊗_{Z} F_{∗}]_{Zπ}^{op} where U(g) sends h ⊗ x to hg ⊗ u(g^{−1})(x) for g ∈ π ,h ∈ π and x ∈ F_{∗}. These data fit together giving a symmetric chain representation (Zπ⊗_{Z}F_{∗}, α, U) of Zπ with the w-twisted involution into D_{1}(Zπ) with the (−1)^{k} · vw-twisted involution. We have introduced in 1.1 the associated transfer homomorphism from L_{n}(Zπ, w) to L_{n+1}(Zπ, (−1)^{k} · vw) = L_{n+2k+1}(Zπ, vw). We shall show in section 11 that this transfer depends only on the class of (M, ψ) in L^{2k+1}(π, Z, v). This finishes the definition of the pairing 2.4.

The groups L^{d}(π, Z, v) fit into a localization exact sequence of the type studied in Chapter 3 of Ranicki [30]

… → L^{d}(π, Z, v) → L^{d}(π, Q, v) → L^{d}(π, Z, Q, v) → L^{d−1}(π, Z, v) → …,

with L^{d}(π, Z, Q, v) the Witt groups of (π, v)-equivariant linking forms\formations on finite abelian groups, L^{2k}(π, Q, v) the Witt group of nonsingular (π, v)-equivariant (−)^{k}-symmetric forms on finite-dimensional vector spaces over Q, and L^{2k+1}(π, Q, v) = 0.

We consider a fibration F ⟶ E ⟶^{p} B and assume that F is an orientable finite Poincaré complex . Then v̂ factorizes over π_{1}(p) by lemma 1.4. The induced homomorphism v : π = π_{1}(B) ⟶ {±1} maps g to the degree of the automorphism H_{d}(ω(g)) of H_{d}(F) ≅ Z. Suppose that the fibre dimension d is 2k. The intersection form on F

H_{k}(F, Z)/tors (H_{k}(F, Z)) ⊗ H_{k}(F, Z)/tors (H_{k}(F, Z)) ⟶ Z (x, y) ↦ ⟨x ∪ y, [F]⟩

is a nonsingular (−1)^{k}-symmetric form over Z. The fibre transport ω : π_{1}(B) ⟶ [F, F] induces a left Zπ-module structure on it. Thus the intersection form becomes actually a nonsingular (π, v)-equivariant (−1)^{k}-symmetric form over Z and defines an element σ^{∗}(F, ω) in L^{2k}(π, Z, v). Now suppose that the fibre dimension is 2k + 1. The universal coefficient theorem yields an isomorphism

ϕ : tors H_{k}(F, Z) ⟶ hom_{Z}(tors H^{k+1}(F, Z), Q/Z) \overline{x} ↦ (f ↦ \frac{f(y)}{s})

where x ∈ C_{k}(F, Z), y ∈ C_{k+1}(F, Z), s ∈ Z\{0}, f ∈ C^{k+1}(F, Z) and sx = d(y). Poincaré duality gives an isomorphism [F] ∩ − : tors H^{k+1}(F, Z) ⟶ tors H_{k}(F, Z). The nonsingular (−1)^{k+1}-symmetric linking form over Z on F

ψ : tors H_{k}(F, Z) ⟶ hom_{Z}(tors H_{k}(F, Z), Q/Z)

is given by hom_{Z}([F] ∩ −, id) ∘ ϕ. The fibre transport ω : π_{1}(B) ⟶ [F, F] induces a left Zπmodule structure on tors H_{k}(F, Z). Thus the linking form becomes actually a nonsingular (π, v)-equivariant (−1)^{k+1}-symmetric linking form over Z and defines an element σ^{∗}(F, ω) in L^{2k+1}(π, Z, v). All in all we have defined an element

2.5 σ^{∗}(F, ω) ∈ L^{d}(π, Z, v)

depending on the fibre F and the operation of π_{1}(B) on its homology and a pairing

2.6 ⊗ : L^{d}(π, Z, v) ⊗ L_{n}(Zπ, w) ⟶ L_{n+d}(Zπ, vw)

Theorem 2.7 (up-down formula) Let F ⟶ E ⟶^{p} B be a fibration of connected spaces such that F is an orientable finite d-dimensional Poincaré complex. Let w : π ⟶ {±1} be any homomorphism and v : π ⟶ {±1} be defined as above. Then the composition

p_{∗} ∘ p^{∗} : L_{n}(Zπ, w) ⟶ L_{n+d}(Zπ, vw)

is given by σ^{∗}(F, ω)⊗?

The proof of the up-down formula 2.7 is deferred to sections 9 and 10. We consider the special case of an orientable fibration. Recall that the L-groups are 4-periodic, i.e. L_{n}(Zπ, w) = L_{n+4}(Zπ, w). Consider the Witt group L^{d}({1}, Z, 1) of the trivial group {1}. There is a pairing

2.8 ⊗ : L^{d}({1}, Z, 1) ⊗ L_{n}(Zπ, w) ⟶ L_{n+d}(Zπ, w) which is given by the tensor product over Z for d even and by the tensor product over Z of a resolution of the linking form for d odd. The pairings 2.8 and 2.6 are compatible with the canonical homomorphism

res : L^{d}({1}, Z, 1) ⟶ L^{d}(π, Z, v)

induced by restriction with the trivial map π ⟶ {1}. If d is 2 or 3 modulo 4, L^{d}({1}, Z, 1) is trivial. A nonsingular symmetric form (M, ψ) over Z induces a nonsingular symmetric form over R which can be written as a sum of p copies of (R, 1) and q copies of (R, −1). The signature of (M, ψ) is defined to be p − q. If d is divisible by 4, the signature induces an isomorphism from L^{d}({1}, Z, 1) to Z. The pairing 2.8 is given by multiplication with the signature. The deRham invariant of a nonsingular skew-symmetric linking form (M, ψ) over Z is the reduction mod 2 of the number of summands in the decomposition of the finite abelian group M as a direct sum of cyclic subgroups of type Z/p^{s} for p prime and s ≥ 1. Suppose that d is 1 modulo 4. Then the deRham invariant defines an isomorphism from L^{d}({1}, Z, 1) to Z/2 (see Ranicki [30], section 4.3 for more details, but ignore the first of the two definitions on p.418). Define the homomorphism

2.9 ν : L_{n}(Zπ, w) ⟶ L_{n+d}(Zπ, w)

by product with the generator in L^{d}({1}, Z, 1) ≅ Z/2 for d ≡ 1(mod 4).

Corollary 2.10 Let F ⟶ E ⟶^{p} B be a fibration of connected spaces such that F is an orientable finite Poincaré complex. Assume that π acts trivially on H_{k}(F, Z)/tors H_{k}(F, Z), if the fibre dimension is d = 2k, and trivially on tors H_{k}(F, Z), if d = 2k + 1. Then :

1. If d is 2 or 3 modulo 4, then p_{∗} ∘ p^{∗} vanishes.

2. If d is divisible by 4, p_{∗} ∘ p^{∗} is multiplication by the signature of F.

3. Suppose d is 1 modulo 4. Then p_{∗} ∘ p^{∗} is given by the map ν defined above, if the deRham invariant of F is non-trivial, and is zero otherwise. In particular 2 · p_{∗} ∘ p^{∗} is zero.

\underline{\underline{Proof}} : Under the assumptions above F defines element σ^{∗}(F) ∈ L^{d}(\{1\}, Z, 1) whose image under res is just σ^{∗}(F, ω). Now the claim follows from the remarks above and naturality.

Example 2.11 Let F ⟶ E ⟶^{p} B be a fibration of connected spaces with the d-dimen-sional sphere S^{d} as fibre. Suppose d ≥ 2. Then p is the boundary of a D^{d+1}-fibration with π_{1}(S^{d}) = π_{1}(D^{d+1}), and the surgery transfer is always zero. This is not true for the K-theory transfer on K_{0} and K_{1} (see Anderson [2]). On the other hand, if the fibration p is a product bundle with S^{1} as fibre, the K-theory transfer on both K_{0} and K_{1} is trivial because of the product formulas for the finiteness obstruction and Whitehead torsion, whereas the surgery transfer is injective modulo 2-torsion by the splitting theorems and Rothenberg sequences of Shaneson [33] and Ranicki [28]. See the appendix to Munkholm and Ranicki [24] for the connection between the S^{1}-bundle transfer in L-theory and the duality in K-theory.

Remark 2.12 For finite π the pring 2.4 is already defined in Yoshida [39]. Yoshida needs the finiteness of π as he has to deal with Zπ-resolutions instead of Z-resolutions with a homotopy π-action. Let N be a closed manifold with π-action. Consider the fibre bundle p : B ×_{π} N ⟶ B with fibre N if π = π_{1}(B). The application of the up-down formula to p gives the main result in Yoshida [39] without the assumption that π is finite.

3. Symmetric Signature

If one is willing to invert 2, one can compute surgery obstructions as the difference of symmetric signatures. Let B be a finite n-dimensional Poincaré complex. In Ranicki [30], section 1.2 the symmetric L-group L^{n}(Zπ_{1}(B), w(B)) and the symmetric signature of B

3.1 σ^{∗}(B) ∈ L^{n}(Zπ_{1}(B), w(B))

are defined. The symmetrization map (1 + T) : L_{n}(Zπ_{1}(B), w(B)) ⟶ L^{n}(Zπ_{1}(B), w(B)) is an isomorphism modulo 8-torsion and sends the surgery obstruction of a normal map f : M ⟶ B with B as target to the difference σ^{∗}(M) − σ^{∗}(B) of the symmetric signatures, if B and M are closed. Product with the generator E_{8} ∈ L_{0}(Z) = Z defines a map E_{8} : L^{n}(Zπ_{1}(B), w(B)) ⟶ L_{n}(Zπ_{1}(B), w(B)) such that E_{8} ∘ (1 + T) = 8 · id : L^{n}(Zπ_{1}(B), w(B)) ⟶ L^{n}(Zπ_{1}(B), w(B)). Note that our construction of a transfer map on the quadratic L-groups need not extend to the symmetric L-groups (see Lück-Ranicki [18], appendix 2). In even dimensions there is also a pairing on the symmetric L-groups given by the tensor product over Z and the diagonal operation which is compatible with the symmetrization map :

3.2 ⊗ : L^{2k}(π, Z, v) ⊗ L^{n}(Zπ, w) ⟶ L^{n+2k}(Zπ, vw)

The details of this pairing are just as in the quadratic case, and are therefore omitted.

Theorem 3.3 Let F ⟶ E ⟶^{p} B be a smooth fibre bundle of connected closed manifolds. If the fibre dimension d is odd we have:

128 · p_{∗}σ^{∗}(E) = 0 ∈ L^{n+d}(Zπ, vw) .

If d is even we have:

8 · p_{∗}σ^{∗}(E) = 8 · σ^{∗}(F, ω) ⊗ σ^{∗}(B) ∈ L^{n+d}(Zπ, vw)

\underline{\underline{Proof}} : Suppose that d is even. Let f : M ⟶ N be a surgery problem of simply-connected 4l-dimensional manifolds whose surgery obstruction is the generator in L_{4l}(Z) ≅ Z. Consider the fibre bundle p × id_{N} : E × N ⟶ B × N. The following diagram commutes by the updown formula 2.7, where π = π_{1}(B), Γ = π_{1}(E) and w = w(B).

L_{n+4l}(Zπ_{1}(B × N), w(B × N)) ⟶ L_{n}(Zπ, w) ⟶ L^{n}(Zπ, w)
↓ (p × id_{N})^{∗}  ↓ p^{∗}
L_{n+4l+d}(Zπ_{1}(E × N), w(E × N)) ⟶ L_{m+d}(ZΓ, ŵv̂)  ↓ σ^{∗}(F, ω)⊗?
↓ (p × id_{N})_{∗}  ↓ p_{∗}
L_{n+4l+d}(Zπ_{1}(B × N), w(B × N)) ⟶ L_{n+d}(Zπ, w) ⟶ L^{n+d}(Zπ, vw)

The left horizontal arrows are given by induction with the projection to the first factor and the two right horizontal maps are symmetrization homomorphisms. The surgery obstruction of the normal map id_{B} × f : B × M ⟶ B × N lies in the left upper corner. Its image under the clockwise composition in the right lower corner is 8 · σ^{∗}(F, ω) ⊗ σ^{∗}(B) because the product formula for surgery obstructions (see Ranicki [30], section 1.9) and the fact that the symmetrization map from L_{4}(Z, 1) to L^{4}(Z, 1) sends the generator to the symmetric form of signature 8. For the same reason its image under the anti-clockwise composition is 8 · p_{∗}σ^{∗}(E).

If d is odd, p_{∗} ∘ p^{∗} : L_{n}(Qπ, w) ⟶ L_{n+d}(Qπ, vw) is zero. This follows from theorem 2.7, since a nonsingular linking form vanishes rationally or from the proof of theorem 4.3. Now a similar argument as above shows that the image of 8 · σ^{∗}(E) under the change of coefficients homomorphism L^{n+d}(Zπ, vw) ⟶ L^{n+d}(Qπ, vw) is zero. The kernel of this map is 16-torsion.

The signature of a nonsingular symmetric form over Z induces an isomorphism from L^{4k}(Z, 1) to Z. Its composition with the induction homomorphism associated to the trivial map pr : π_{1}(B) ⟶ {1} sends σ^{∗}(B) to the signature sign(B). Note for the trivial group that symmetric L-group L^{d}(Z, 1) can be identified with Witt group L^{d}({1}, Z, 1) and the corresponding pairings 2.6 and 3.2 agree. We get from an argument similar to the one used in the proof of theorem 3.3 and corollary 2.10

Corollary 3.4 Let F ⟶ E ⟶^{p} B be an smooth fibre bundle of connected closed manifolds. Suppose that p is orientable, i.e. ω : π_{1}(B) ⟶ [F, F] is trivial. Then :

1. If the dimension d ≡ 1(mod 4), then :

16 · p_{∗}σ^{∗}(E) = 0 ∈ L^{n+d}(Zπ, vw) .

If the dimension d ≡ 2, 3(mod 4), then :

8 · p_{∗}σ^{∗}(E) = 0 ∈ L^{n+d}(Zπ, vw) .

2. If the dimension of d is divisible by 4, then :

8 · p_{∗}σ^{∗}(E) = 8 · sign(F) · σ^{∗}(B) ∈ L^{n+d}(Zπ, vw)

3. Suppose that F, B and E are orientable. If their dimensions are all divisible by 4, then

:

sign(E) = sign(F) · sign(B) ∈ Z .

Otherwise sign(E) = 0 .

The last statement in corollary 3.4 is a result due to Chern, Hirzebruch and Serre [6]. The examples of Kodaira [13], Atiyah [3], Hirzebruch [12] of fibre bundles in which the signature is not multiplicative show that some condition such as orientability is necessary for the signature to be multiplicative.

Let d be 2k and π be π_{1}(B) and assume that n + d is divisible by 4. The structure ψ : H_{k}(F, Q) ⟶ hom_{Q}(H_{k}(F, Q), Q) of a nonsingular (π, 1)-equivariant (−1)^{k}-symmetric form over Q on H_{k}(F, Q) is induced by the fibre transport, Poincaré duality and the universal coefficient theorem. Let σ^{∗}(B)∩? : hom_{Qπ}(C_{n−∗}(B̃, Q), Qπ) ⟶ C_{∗}(B̃) be the Poincaré Qπ-chain equivalence. The following composition is a Qπ-chain equivalence

hom_{Q}(C_{n−∗}(B̃, Q) ⊗_{Qπ} H_{k}(F, Q), Q) ⟶^{hom(id⊗ψ,id)}

hom_{Q}(C_{n−∗}(B̃, Q) ⊗_{Qπ} hom_{Q}(H_{k}(F, Q), Q), Q)

⟶^{ρ} hom_{Qπ}(C_{n−∗}(B̃, Q), Qπ) ⊗_{Qπ} H_{k}(F, Q) ⟶^{σ^{∗}(B)⊗id∩?} C_{∗}(B̃, Q) ⊗_{Qπ} H_{k}(F, Q)

where ρ is the canonical isomorphism. The composition of the inverse of the isomorphism on H_{n/2} induced by the chain map above with the canonical isomorphism from hom_{Q}(H_{n/2}(C_{∗}(B̃) ⊗_{Qπ} H_{k}(F, Q), Q) to hom_{Q}(H_{n/2}(B, H_{k}(F, Q)), Q) determines the structure ϕ : H_{n/2}(B, H(F, Q)) ⟶ hom_{Q}(H_{n/2}(B, H_{k}(F, Q)), Q) of a nonsingular symmetric form over Q. We derive from theorem 3.3:

Corollary 3.5 Let F ⟶ E ⟶^{p} B be a smooth fibration of connected closed manifolds such that d + n is divisible by 4. Then the signature sign(E) is zero if d is odd. Suppose that d is 2k. Then we have :

sign(E) = sign ∘ pr_{∗}(σ^{∗}(F, ω) ⊗ σ^{∗}(B)) = sign(H_{n/2}(B, H_{k}(F, Q)), ϕ)

where pr : π ⟶ {1} is the trivial map.

Corollary 3.5 was first proved by Meyer [20], Satz I 2.2.

See the appendix for a discussion of the relation between our results and the characteristic class formulae relating the non-multiplicativity of the signature in a fibre bundle F ⟶ E ⟶^{p} B to the action of π_{1}(B) on H^{∗}(F; R).

4. Rational Computations

If one allows rational coefficients instead of integral ones, computations simplify. The algebraic transfer of a fibration is also defined for rational coefficients and is compatible with the change of coefficients homomorphisms. The change of coefficients homomorphism L_{n}(Zπ, w) ⟶ L_{n}(Qπ, w) is an isomorphism if one inverts 2 (see Ranicki [30], page 376). Also the difference between symmetric and quadratic L-theory and the difference between any type of decorated L-groups like L^{h}, L^{s} and L^{p} vanishes, when inverting 2. Hence we obtain a vanishing result for the integral surgery transfer (for any decoration), when 2 is inverted, if we prove the vanishing of the rational algebraic surgery transfer p^{∗} : L_{n}^{p}(Qπ_{1}(B), w) ⟶ L_{n+d}^{p}(Qπ_{1}(E), v̂ŵ) for the projective L-groups. Recall that L^{p} means that we allow finitely generated projective modules instead of finitely generated free ones.

Let F ⟶ E ⟶^{p} B be a fibration of connected spaces such that F is a finitely dominated Poincaré complex. Finitely dominated means that F is up to homotopy a retract of a finite CW complex. We abbreviate π = π_{1}(B), Γ = π_{1}(E), and write the kernel of π_{1}(p) : Γ ⟶ π by Δ. Let w : π ⟶ {±1} be any homomorphism. Associated with these data is the rational surgery transfer p^{∗} : L_{n}^{p}(Qπ, w) ⟶ L_{n+d}^{p}(QΓ, v̂ŵ). Now suppose that Δ is finite. Then restriction with the epimorphism π_{1}(p) defines a homomorphism

4.1 res : L_{n}^{p}(Qπ, w) ⟶ L_{n}^{p}(QΓ, ŵ)

where ŵ was defined to be w ∘ π_{1}(p) in 1.3. Let L^{2k}(Γ, Q, v) be the Witt group of stable isomorphism classes of (Γ, v)-equivariant (−1)^{k}-symmetric forms over Q, where v̂ was introduced in 1.3. The tensor product over Q with the diagonal γ-action gives a pairing

4.2 ⊗ : L^{2k}(Γ, Q, v) ⊗ L_{n}(QΓ, w) ⟶ L_{n+2k}(QΓ, vw)

Let \overline{F} be the covering of F associated to π_{1}(F) ⟶ Δ. The pointed fibre transport and the intersection pairing induce the structure of a nonsingular (Γ, v̂)-equivariant (−1)^{k}-symmetric form over Q on H_{k}(F, Q), We get a class σ^{∗}(\overline{F} , ω^{+}) in L^{2k}(Γ, Q, v).

Theorem 4.3 Let F ⟶ E ⟶^{p} B be a fibration of connected spaces such that F is a finitely dominated Poincaré complex and the kernel Δ of π_{1}(p) is finite. Then

p^{∗} : L_{n}^{p}(Qπ, w) ⟶ L_{n+d}^{p}(QΓ, v̂ŵ)

is zero if the fibre dimension d is odd, and is the composition (σ^{∗}(\overline{F} , ω^{+})⊗?) ∘ res, if d is 2k.

\underline{\underline{Proof}} : The rational surgery transfer is given by a symmetric representation of Qπ in D_{d}(QΓ) (see 1.5 and lemma 1.4)

U : Qπ ⟶ [QΓ ⊗_{QΔ} C(\overline{F} , Q), QΓ ⊗_{QΔ} C(\overline{F} , Q)]_{QΓ}^{op}

Regard the rational homology QΓ⊗_{QΔ} H(\overline{F} , Q) as a d-dimensional QΓ-chain complex using the trivial differential. There is a QΔ-chain equivalence i : QΓ ⊗_{QΔ} H(\overline{F} , Q) ⟶ C(\overline{F} , Q) uniquely determined up to homotopy by the property that it induces the identity on homology. The QΔ-chain isomorphism α : hom_{Q}(H_{d−∗}(\overline{F} , Q), Q) ⟶ QΓ ⊗_{QΔ} H(\overline{F} , Q) is given by Poincaré duality and the natural isomorphism hom_{Q}(H_{d−∗}(\overline{F} , Q), Q) ⟶ H^{d−∗}(\overline{F} , Q). The pointed fibre transport induces a QΓ-structure on QΓ ⊗_{QΔ} H(\overline{F} , Q). We put the diagonal QΓ-structure on QΓ ⊗_{QΔ} H(\overline{F} , Q). Because α extends to a QΓ-chain isomorphism β from hom_{QΓ}(QΓ ⊗_{QΔ} H(\overline{F} , Q), QΓ) to QΓ ⊗_{QΔ} H(\overline{F} , Q), we obtain a symmetric representation of Qπ in D_{d}(QΓ)

V : Qπ ⟶ [QΓ ⊗_{QΔ} H(\overline{F} , Q), QΓ ⊗_{QΔ} H(\overline{F} , Q)]_{QΓ}^{op}

where V (g) sends h ⊗ x to hg ⊗ x. Define a QΓ-chain map from Qπ ⊗_{Q} H_{∗}(\overline{F} , Q) to QΓ ⊗_{QΔ} H(\overline{F} , Q) by mapping g ⊗ x to g̃ ⊗ g̃^{−1}x for any lift g̃ ∈ Γ of g ∈ π. Composing this with QΓ ⊗_{QΔ} i defines a QΓ-chain equivalence

j : QΓ ⊗_{QΔ} H(\overline{F} , Q) ⟶ QΓ ⊗_{QΔ} C(\overline{F} , Q)

One easily checks that j is an isomorphism of symmetric representations of Qπ in D_{d}(QΓ). Hence the surgery transfer associated with (QΓ ⊗_{QΔ} H(\overline{F} , Q), β, V) is the rational surgery transfer p^{∗}. We get from the definitions that p^{∗} is the composition

L_{n}(Qπ, w) \underrightarrow{res} L_{n}(QΓ, ŵ) \underrightarrow{⊗[H(\overline{F} , Q), α]} L_{n+d}(QΓ, v̂ŵ)

where the last map is given by the tensor product over Q and the diagonal Γ-action. It remains to show that the second map is zero, if d is odd, and σ^{∗}(\overline{F} , ω^{+})⊗? if d is even.

Let D be the (k − 1)- resp. k-dimensional QΓ-chain complex obtained by truncating H(\overline{F} , Q) if d is 2k resp. 2k + 1. Let p : H(\overline{F} , Q) ⟶ D be the canonical projection. Choose for λ ∈ L_{n}(QΓ, ŵ) a n-dimensional quadratic Poincaré QΓ-chain complex (C, \{ψ_{s}\}) representing λ. The image of λ under [H(\overline{F} , Q), α]⊗? is represented by a (n + d)- dimensional quadratic Poincaré QΓ-chain complex whose underlying QΓ-chain complex is C ⊗_{Q} H(\overline{F} , Q). Now we can do algebraic surgery on id ⊗ p : C ⊗_{Q} H(\overline{F} , Q) ⟶ C ⊗ D. The result is contractible if d is odd, and homotopy equivalent to the obvious representative of σ^{∗}(\overline{F} , ω^{+}) ⊗ λ if d is even.

5. Untwisted Fibrations

A fibration F ⟶ E ⟶^{p} B is called untwisted, if the pointed fibre transport ω^{+} : π_{1}(E) ⟶ [F, F,]^{+} is trivial, and orientable if the fibre transport ω : π_{1}(B) ⟶ [F, F] is trivial. Let G_{1}(F) be the kernel of τ : π_{1}(F) ⟶ [F, F]^{+} as defined by Gottlieb [10]. A fibration is untwisted if and only if it is orientable and G_{1}(F) = π_{1}(F) (see lemma 1.4). Because G_{1}(F) = π_{1}(F) holds for H-spaces, an orientable fibration with H-space as fibre is untwisted. Untwisted fibrations seem to be the largest class of fibrations where one can hope to prove the vanishing of the surgery transfer when one has only little information about the fibre.

Let F ⟶ E ⟶^{p} B be an untwisted fibration of connected spaces such that the fibre F is a d-dimensional finite Poincaré complex. We abbreviate π = π_{1}(B) and Γ = π_{1}(E) and write Δ for the kernel of π_{1}(p). Consider a homomorphism w : π = π_{1}(B) ⟶ {±}. Put ŵ to be w ∘ π_{1}(p). Since p is untwisted, Δ is central. The tensor product over ZΔ induces a pairing

5.1 ⊗_{ZΔ} : L^{d}(ZΔ, 1) ⊗ L_{n}(ZΓ, ŵ) ⟶ L_{d+n}(ZΓ, ŵ)

If Δ is trivial, the pairing 5.1 above reduces to the pairing 2.8. Let σ^{∗}(\overline{F}) ∈ L^{d}(ZΔ, 1) be the symmetric signature of the covering \overline{F} of F associated with π_{1}(F) ⟶ Δ. The following theorem is proven for the K-theory transfer in Lück [15] page 165. The L-theoretic version is proven similarly.

Theorem 5.2 (down-up formula for untwisted fibrations) If the conditions above are satisfied, the composition

p^{∗} ∘ p_{∗} : L_{n}(ZΓ, ŵ) ⟶ L_{n+d}(ZΓ, ŵ)

is given by σ^{∗}(\overline{F})⊗_{ZΔ}?.

Let F be a connected CW complex such that π_{1}(F) = Z × G and Z ⊂ G_{1}(F). Let q : F′ ⟶ F be the infinite cyclic covering associated with the projection π_{1}(F) ⟶ G. Choose a representative i : S^{1} ⟶ F of the generator of Z. Then q ∨ i extends to a homotopy equivalence f : F′ × S^{1} ⟶ F by Lück [15] page 154. We identify F and F′ × S^{1}. If F is a d-dimensional finitely dominated Poincaré complex, then F′ is a (d − 1)-dimensional finitely dominated Poincaré complex.

Let F ⟶ E ⟶^{p} B be an untwisted fibration of connected spaces such that F is a d-dimensional finitely dominated Poincaré complex. We have the central extension with a free abelian kernel Δ′ ⟶ Γ′ ⟶ π if Δ′ is Δ/tors (Δ) and Γ′ is Γ/tors (Δ). Let r be the rank of Δ′. Associated to any such extension is a surgery transfer

5.3 trf : L_{n}^{ϵ}(Rπ, w) ⟶ L_{n+r}^{ϵ}(RΓ′, ŵ)

where ϵ is one of the decorations p, h or s and Z ⊂ R ⊂ Q. It is the surgery transfer associated to the following symmetric representatio of Rπ in D_{r}(RΓ′). Choose an identification Δ′ = Z^{r}. Let C be the symmetric Poincaré ZΔ′-chain complex of the universal covering of the rdimensional torus T^{r}. The symmetric representation is given by 5.4 id ⊗_{ZΔ′} α : RΓ′ ⊗_{ZΔ′} C^{d−∗} ⟶ RΓ′ ⊗_{ZΔ′} C ,

U : Rπ ⟶ [RΓ ⊗_{ZΔ′} C, RΓ ⊗_{ZΔ′} C]_{ZΓ′}^{op}

where we have identified hom_{RΓ′}(RΓ′ ⊗_{ZΔ′} C, RΓ′) with RΓ′ ⊗_{ZΔ′} C^{d−∗} and U sends g ∈ π to h ⊗ x ↦ hg̃ ⊗ x for any lift g̃ ∈ Γ′ of g. Let trf be the associated surgery transfer 1.1. Assume that |tors (Δ)| is invertible in R. Then restriction with the epimorphism Γ ⟶ Γ′ induces a homomorphism res : L_{n+r}(RΓ′) ⟶ L_{n+r}(RΓ). Let sign(\overline{F}) ∈ Z be the signature of the covering \overline{F} of F associated with π_{1}(F) ⟶ Δ. This is well-defined since F̂ is a finitely dominated Poincaré complex. Let ν : L_{n}(Rπ, w) ⟶ L_{n+1}(Rπ, w) be the homomorphism defined in 2.9, but now with coefficients in R.

Theorem 5.5 We get for the surgery transfer

p^{∗} : L_{n}^{p}(Rπ, w) ⟶ L_{n+d}^{p}(RΓ, v̂ŵ)

with coefficients in R under the conditions and in the notation above :

1. p^{∗} is zero if one of the following conditions is satisfied:

(a) The kernel of π_{1}(F) ⟶ π_{1}(E) is infinite

(b) d − r ≡ 2, 3 (4)

(c) d − r ≡ 1 (4) and 1/2 ∈ R

(d) d − r ≡ 1 (4) and the deRham invariant of \overline{F} is zero.

2. If π_{1}(F) ⟶ π_{1}(E) has finite kernel and d − r ≡ 0 (4) then p^{∗} is sign(\overline{F}) · res ∘ trf.

3. Suppose that π_{1}(F) ⟶ π_{1}(E) has finite kernel, 1/2 ∉ R and d − r ≡ 1 (4) and the deRham invariant of \overline{F} is not zero. Then p^{∗} is ν ∘ res ∘ trf.

4. If tors Δ is trivial the items above hold also for p^{∗} : L_{n}^{ϵ}(Zπ, w) ⟶ L_{n+d}^{ϵ}(ZΓ, v̂ŵ) if ϵ is h,p or s and res is the identity.

\underline{\underline{Proof}} : Consider the symmetric representation of π in D_{d}(RΓ) which is obtained by viewing the symmetric representation of π in D_{r}(ZΓ′) in 5.4 as a symmetric representation from π into D_{r}(ZΓ) by restriction with Γ ⟶ Γ′ and tensoring it with C(\overline{F} , R). Because (RΓ′ ⊗_{RΔ′} C(T̃^{r}, R)) ⊗_{R} C(\overline{F} , R) and RΓ ⊗_{Rπ_{1}(_{F})} C(F̃, R) are RΓ-chain equivalent, one obtains an isomorphism from this symmetric representation to the one describing p^{∗}. This implies that the algebraic transfer p^{∗} : L_{n}^{p}(Rπ, w) ⟶ L_{n+d}^{p}(RΓ, v̂ŵ) is the composition

L_{n}^{p}(Rπ, w) \underrightarrow{trf} L_{n+r}^{p}(RΓ′, ŵ) \underrightarrow{res} L_{n+r}^{p}(RΓ, ŵ) ⟶^{σ^{∗}(\overline{F})⊗_{R}?} L_{p}^{n+d}(RΓ, ŵ)

where σ^{∗}(\overline{F}) ∈ L^{d−r}(R, 1) is the symmetric signature of \overline{F} and ⊗_{R} is the analogue of the pairing 2.8.

If d − r ≡ 2, 3 (4), then L^{d−r}(Z, 1) is trivial. The class σ^{∗}(\overline{F}) ∈ L^{d−r}(Z, 1) is detected by the signature if d − r ≡ 0 (4), and by the deRham invariant if d − r ≡ 1 (4). Suppose that the kernel of π_{1}(F) ⟶ π_{1}(E) is infinite. As G_{1}(\overline{F}) = π_{1}(\overline{F}) and the rank of π_{1}(\overline{F}) and the kernel above agree, \overline{F} is homotopy equivalent to S^{1} × Y for some Y. As S^{1} × Y bounds D^{2} × Y, the class σ^{∗}(\overline{F}) ∈ L^{d−r}(Z, 1) is trivial. If d − r ≡ 1 (4) and 1/2 ∈ R, σ^{∗}(\overline{F}) ∈ L^{d−r}(R, 1) is trivial. This shows item 1.), 2.) and 3.). One obtains 4.) similarly, using the fact that \overline{F} is a finite simple Poincaré complex, as finiteness obstructions and Whitehead torsions are always trivial over the principal domain Z.

Recall from example 2.11 that the surgery transfer of the product bundle with a torus as fibre is injective modulo 2-torsion so that the vanishing results above are the best we can expect.

6. G-Principal Bundles

Corollary 6.1 Let G be a compact connected Lie group and G ⟶ E ⟶^{p} B be a G-principal bundle.

1. If G is not a torus, then

p^{∗} : L_{n}(Zπ, w) ⟶ L_{n+d}(ZΓ, v̂ŵ)

is trivial.

2. Suppose that G is not a torus or that π_{1}(G) ⟶ π_{1}(E) is not injective. Then

p^{∗} : L_{n}^{p}(Qπ_{1}(B), w) ⟶ L_{n+d}^{p}(Qπ_{1}(E), v̂ŵ)

is trivial.

\underline{\underline{Proof}} : 1.) Since G is non-abelian, G contains a non-abelian 3-dimensional Lie subgroup H. Let S^{1} be the maximal torus in H. We can write the given G-principal bundle as the composition

p : E ⟶^{p_{1}} E/S^{1} ⟶^{p_{2}} E/H ⟶^{p_{3}} B

It is obvious from the definition of the geometric transfer that p^{∗} is the composition p_{1}^{∗}∘p_{2}^{∗}∘p_{3}^{∗}. Hence it suffices to show that p_{2}^{∗} is zero. As the fibre of p_{2} is S^{2}, the claim follows from the up-down formula 2.7, or else by 2.11.

2.) follows from theorem 5.5

Recall from example 2.11 that the transfer of the trivial bundles with fibre S^{1} is nontrivial so that the vanishing results above are optimal.

7. Change of K-theory

So far we have mainly dealt with L^{h} in the integral and L^{p} in the rational case. We make some comments what happens for arbitrary decorations. Consider a fibration F ⟶ E ⟶^{p} B of connected spaces such that F is a finitely dominated d-dimensional Poincaré complex. Let w : π = π_{1}(B) ⟶ {±1} be a homomorphism. Let X resp. Y be a subgroup of K_{i}(Zπ_{1}(B)) resp. K_{i}(Zπ_{1}(E)) containing the image of K_{i}(Z) and closed under the w-twisted resp. v̂ŵ-twisted involution (given by taking dual modules and maps) for i = 0, 1. Then we have the decorated L-group L_{n}^{X}(Zπ_{1}(B), w) resp. L_{n}^{Y} (Zπ_{1}(E), v̂ŵ). In order to get a symmetric chain representation from the fibration and hence a well-defined algebraic transfer, we have to specify a base point b ∈ B, a CW complex G and a homotopy equivalence f : G ⟶ F_{b}. If i is 0, we assume that the image of the finiteness obstruction o(G) of G under (j ∘ f)_{∗} : K_{0}(Zπ_{1}(G)) ⟶ K_{0}(Zπ_{1}(E)) for the inclusion j : F_{b} ⟶ E lies in Y. If i is 1, we assume that G is a finite Poincaré complex and the image of the Whitehead torsion τ([G] ∩ −) of the Poincaré chain equivalence [G] ∩ − : C(G̃)^{d−∗} ⟶ C(G̃) under (j ∘ f)_{∗} : Wh(π_{1}(G)) ⟶ Wh(π_{1}(E)) lies in Y. Moreover, we assume that the K-theory transfer p^{∗} : K_{i}(Zπ_{1}(B)) ⟶ K_{i}(Zπ_{1}(E)) maps X to Y. Under these assumptions there is is an algebraic surgery transfer

p^{∗} : L_{n}^{X}(Zπ_{1}(B), w(B)) ⟶ L_{n+d}^{Y}(Zπ_{1}(E), v̂ŵ) .

It is easy to check in the case i = 0 that (j ∘ f)_{∗}(o(G)) and the surgery transfer p^{∗} do not depend on the choice of b, G and f. This is also true for L^{h}. In the simple case this choice can matter. If p is a smooth bundle of compact manifolds and f : G ⟶ F_{b} is given by a triangulation, then these choices do not matter and o(G) and τ([G] ∩ −) are trivial.

Let Y′ be the image of Y under p_{∗} : K_{i}(Zπ_{1}(E)) ⟶ K_{i}(Zπ) for π = π_{1}(B). Then the composition

p_{∗} ∘ p^{∗} : L_{n}^{X}(Zπ, w) ⟶ L_{n+d}^{Y′}(Zπ, vw)

is defined. Let Z be an abelian subgroup of Sw(π, Z) closed under the v-twisted involution. Suppose that the pairing 2.1 sends X ⊗ Z to Y′. Let L_{Z}^{d}(π, Z, v) be the decorated analogue of L^{d}(π, Z, v), in which it is demanded additionally that the classes of all the Zπ-modules lie in Z ⊆ Sw(π, Z). We make the following assumption :

7.1 ∑_{i=0}^{k−1} (−1)^{i} · [H_{i}(F)] and ∑_{i=0}^{d} (−1)^{i} · [H_{i}(F)] lie in Z ⊂ Sw(π, Z) if d = 2k

[H_{k}(F)/tors H_{k}(F)] + ∑_{i=0}^{k−1} (−1)^{i} · [H_{i}(F)] and ∑_{i=0}^{d} (−1)^{i} · [H_{i}(F)] lie in Z ⊂ Sw(π, Z) if d = 2k + 1

We shall explain at the end of section 10 that the definitions of 2.5 and 2.6 carry over so that one gets a class

σ^{∗}(F, ω) ∈ L_{Z}^{d}(π, Z, v)

and a pairing

⊗ : L_{Z}^{d}(π, Z, v) ⊗ L_{n}^{X}(Zπ, w) ⟶ L_{n+d}^{Y′}(Zπ, vw)

and the up-down formula 2.7 is verified, i.e. p_{∗} ∘ p^{∗} is given by σ^{∗}(F, ω)⊗?.

Let G be a connected compact Lie group and G ⟶ E ⟶^{p} B be a G-principal bundle. Suppose that G is not a torus. Let p_{1} ∘ p_{2} ∘ p_{3} be the decomposition of p appearing in the proof of theorem 6.1. Then the K-theory transfer p^{∗} is 2 · p_{1}^{∗} ∘ p_{3}^{∗} since the fibre S^{2} of p_{2} is simply connected and has Euler characteristic 2. Suppose that Y contains not only the image of X under the K-theory transfer p^{∗}, but also the image of X under p_{1}^{∗} ∘ p_{3}^{∗}. Then the argument in theorem 6.1 shows that the surgery transfer p^{∗} : L_{n}^{X}(Zπ_{1}(B), w(B)) ⟶ L_{n+d}^{Y}(Zπ_{1}(E), v̂ŵ) vanishes.

8. Review of Construction of Algebraic Surgery Transfer

Let R and S be rings with involution and (C, α, U) be a symmetric representation of R into D_{d}(S) as introduced in section 2. In this section we recall the construction of the algebraic surgery transfer (C, α, U)^{∗} : L_{n}(R) ⟶ L_{n+d}(S) as defined in Lück-Ranicki [18].

An involution on an additive category A is a contravariant functor ∗ : A ⟶ A together with a natural equivalence e : id_{A} ⟶ ∗ ∘ ∗ such that e(M)^{∗} and (e(M)^{−1})^{∗} agree for all objects M in A. An example is the additive category B(R) of finitely generated based free Rmodules with the involution sending a based R-module M to the dual R-modul hom_{R}(M, R) with the dual base. The additive category D_{d}(S) of finitely generated free d-dimensional Schain complexes with homotopy classes of chain maps as morphisms possesses the involution sending C to the dual chain complex C^{d−∗}. A symmetric representation (C, α, U) of R in D_{d}(S) extends in a unique way to a functor of additive categories with involutions, also denoted by (C, α, U) : B(R) ⟶ D_{d}(S).

The definition of the quadratic L-groups in terms of forms and formations as well in terms of quadratic Poincaré complexes for modules over a ring with involution carries over directly to additive categories with involution (see Ranicki [31]). In even dimensions an element in L_{2m}(D_{d}(S)) is represented by a nonsingular (−1)^{m}-quadratic form (C, [ψ]), i.e., an object C together with a homotopy class [ψ] of chain maps ψ : C^{d−∗} ⟶ C such that ψ + (−1)^{m} · ψ^{d−∗} is a chain homotopy equivalence C^{d−∗} ⟶ C. In odd dimensions an element in L_{2m+1}(D_{d}(S)) is representated by a nonsingular (−1)^{m}-quadratic formation (C, D, [μ], [γ]), i.e. objects C and D in D_{d}(S) and homotopy classes of chain maps [μ] and [γ] for μ : C ⟶ D and γ : D^{d−∗} ⟶ C such that there exists a chain map θ : D^{d−∗} ⟶ D and a chain homotopy χ : θ − (−1)^{m} · \overline{θ} ≃ μ ∘ γ with the property that cone(−μ^{d−∗}) ⟶ cone(μ)_{∗} given by

( \begin{matrix}0 & γ \\ (−1)^{m}γ^{d−∗} & χ + (−1)^{m} · χ^{d−∗} \end{matrix} ) : C^{d−r} ⊕ D^{d+1−r} ⟶ C_{r−1} ⊕ D_{r}

is a chain equivalence. Note that χ and θ are required to exist but are not part of the structure. This definition of form resp. formation corresponds the notion of a 0- resp. 1dimensional (−1)^{m}-quadratic Poincaré complex in the category D_{d}(S). We put L_{n+4l}(D_{d}(S)) to be L_{n}(D_{d}(S)) for 0 ≤ n ≤ 3 and l ≥ 0.

The symmetric representation (C, α, U) extends uniquely to a functor of additive categories with involution, also denoted by (C, α, U) : B(R) ⟶ D_{d}(S). A functor of additive categories with involutions induces a map on the (quadratic) L-groups. Hence we obtain a homomorphism :

8.1 (C, α, U)⊗? : L_{n}(R) ⟶ L_{n}(D_{d}(S))

The main ingredient in the construction of the surgery transfer is the generalized Morita homomorphism

8.2 μ : L_{n}(D_{d}(S)) ⟶ L_{d+n}(S)

defined as follows. We begin with the case where n is even, n = 2m. Represent the element λ ∈ L_{n}(D_{d}(S)) by a nonsingular (−1)^{m}-quadratic form (C, [ψ]). Choose a representative ψ of [ψ]. Define a d-dimensional (−1)^{m}-quadratic structure {ψ_{s}} on C by ψ_{0} = ψ and ψ_{s} = 0 for s ≥ 1. Then (C, {ψ_{s}}) is a d-dimensional (−1)^{m}-quadratic Poincaré complex in the sense of Ranicki [30], section 1.2. Its class in L_{n+d}(S) = L_{d}(S, (−1)^{m}) is defined to be μ(λ).

Suppose n is odd, n = 2m+1. Represent λ ∈ L_{n}(S) by a nonsingular (−1)^{m}-quadratic formation (C, D, [μ], [γ]). Then μ(λ) ∈ L_{d+1}(S, (−1)^{m}) is represented by the following (d+1)- dimensional (−1)^{m}-quadratic Poincaré complex. The underlying (d+1)-dimensional S-chain complex is the mapping cone of μ : C ⟶ D given by:

8.3 ( \begin{matrix}−c & 0 \\ μ & d \end{matrix} ) : cone(μ)_{r} = C_{r−1} ⊕ D_{r} ⟶ cone(μ)_{−1} = C_{r−2} ⊕ D_{r−1}

The (−1)^{m}-quadratic structure is defined by:

8.4

ψ_{0} = ( \begin{matrix}0 & γ \\ 0 & χ \end{matrix} ) : cone(−μ^{d−∗})_{r} = C^{d−r} ⊕ D^{d+1−r} ⟶ cone(μ)_{r} = C_{r−1} ⊕ D_{r}

ψ_{1} = ( \begin{matrix}0 & 0 \\ 0 & θ \end{matrix} ) : cone(−μ^{d−∗})_{r} = C^{d−r} ⊕ D^{d+1−r} ⟶ cone(μ)_{r−1} = C_{r−2} ⊕ D_{r−1}

ψ_{s} = 0 for s ≥ 2

Define the algebraic surgery transfer associated with the symmetric representation (C, α, U) of R into D_{d}(S)

(C, α, U)^{∗} : L_{n}(R) ⟶ L_{n+d}(S)

to be the composition μ ∘ ((C, α, U)⊗?) of the homomorphisms defined in 8.2 and 8.1.

We make some remarks on sign conventions as they differ from those used in Lück-Ranicki [18]. The mapping cone of a chain map was defined in 8.3. This convention has the property that the cellular chain complex of the (geometric) mapping cone of a cellular map is the (algebraic) mapping cone of the induced chain map. Define the suspension of a chain map C to be the mapping cone of C ⟶ {0}. If the following square

C \underrightarrow{u} C′
f ↓  _{↓ f′}
D \underrightarrow{v} D′

commutes up to the homotopy h : v ∘ f ≃ f′ ∘ u we obtain a chain map

8.5 ( \begin{matrix}u & 0 \\ h & v \end{matrix} ) : cone(f) ⟶ cone(f′)

The suspension ΣC is the mapping cone of C ⟶ {0}. Let C^{−∗} be the chain complex given by (C^{−∗})_{r} = (C_{−r})^{∗} and (c^{−∗})_{r} = (c_{−r+1})^{∗}. Let C^{d−∗} be Σ^{d}C^{−∗}. Define the dual chain map f^{d−∗} : C^{d−∗} ⟶ D^{d−∗} by (f^{d−∗})_{r} = (f_{d−r})^{∗}. One easily checks that with these sign conventions cone(f)^{d−∗} = cone((−1)^{d+1} · f^{d−∗}).

Note that the (−1)^{m}-quadratic structure 8.4 is not directly a (−1)^{m}-quadratic structure in the sense of Ranicki [30], section 1.2. It is a (−1)^{m}-quadratic structure in the following sense. A (−1)^{m}-quadratic structure on the mapping cone of μ is a collection {ψ_{s}} of maps ψ_{s} : cone(−μ^{d−∗}) ⟶ cone(μ) of graded modules of degree −s such that the following relations hold for s ≥ 0

cψ_{s} − (−1)^{s} · ψ_{s}c^{d−∗} = ψ_{s+1} − (−1)^{m} · ψ_{s+1}^{d−∗}

where c is the differential of cone(μ) and c^{d−∗} the differential of cone(−μ^{d−∗}). It is a Poincaré structure if the chain map ψ_{0} + (−1)^{m}ψ^{d−∗} : cone(−μ^{d−∗}) ⟶ cone(μ) is a chain homotopy equivalence. Using the identification cone(f)^{d−∗} = cone((−1)^{d+1} · f^{d−∗}) and the right signs one can associate to such a (−1)^{m}-quadratic Poincaré structure a (−1)^{m}-quadratic Poincaré structure in the sense of Ranicki [30], section 1.2 (cf. Lück-Ranicki [18] proposition 3.2). The necessary signs are cumbersome so that we stick to the definition above and suppress this step in the sequel. This simplifies the notation in the following sections considerably.

9. Proof of the up-down Formula in Even Base Dimensions

This section prepares the proof of the up-down formula 2.7, and completes it for even base dimensions n = 2m.

Let C be a finitely generated free Z-chain complex. A π-twist on C is a choice of Z-chain maps C(g) : C ⟶ C for g ∈ π such that C(1) = id and C(g) ∘ C(h) ≃ C(gh) holds for g, h ∈ π. In particular, we obtain a homomorphism π ⟶ [C, C]_{Z} by sending g to [C(g)] and a Zπ-structure on the homology H(C). A map f : (C, {C(g)}) ⟶ (D, {D(g)}) of chain complexes with a π-twist is a chain map f : C ⟶ D satisfying f ∘ C(g) ≃ D(g) ∘ f for all g ∈ π. Let μ : Zπ^{a} ⟶ Zπ^{b} be a Zπ-homomorphism where Zπ^{a} is the direct sum of a copies of Zπ. Equip Zπ ⊗ C with the Zπ-structure given by u · (v ⊗ x) = uv ⊗ x. In this and in the next section all tensor products are understood to be over the integers Z. We define a Zπ-chain map

9.1 μ ⊗_{t} C : Zπ^{a} ⊗ C ⟶ Zπ^{b} ⊗ C

as follows. As μ can be viewed as a matrix of homomorphism Zπ ⟶ Zπ, it suffices to specify μ ⊗_{t} C in the case a = b = 1. Let μ(1) ∈ Zπ be ∑_{g∈π} λ_{g} · g. Then μ ⊗_{t} C sends u ⊗ x to ∑_{g∈π} λ_{g} · ug ⊗ C(g^{−1})(x). One should not confuse μ ⊗_{t} C with μ ⊗ C which sends g ⊗ x to μ(g) ⊗ x. Let ρ_{2} : Zπ^{a} ⊗ H(C) ⟶ H(Zπ^{a} ⊗ C) be the canonical map. This is an isomorphism as Zπ is Z-free. Define ρ_{1} : Zπ^{a} ⊗ H(C) ⟶ Zπ^{a} ⊗ H(C) by sending g ⊗ x to g ⊗ g^{−1}x. We obtain a Zπ-isomorphism

9.2 ρ = ρ_{2} ∘ ρ_{1} : Zπ^{a} ⊗ H(C) ⟶ H(Zπ^{a} ⊗ C)

if π acts diagonally on the source. We shall use ρ as an identification. One easily checks:

Lemma 9.3 Let μ_{1} : Zπ^{a} ⟶ Zπ^{b} and μ_{2} : Zπ^{b} ⟶ Zπ^{c} be Zπ-homomorphisms. Consider a map of chain complexes with a π-twist f : (C, {C(g)}) ⟶ (D, {D(g)}). Then :

1. (μ_{2} ⊗_{t} C) ∘ (μ_{1} ⊗_{t} C) ≃_{Zπ} (μ_{2} ∘ μ_{1}) ⊗_{t} C

2. (μ_{1} ⊗_{t} D) ∘ (Zπ^{a} ⊗ f) ≃_{Zπ} (Zπ^{b} ⊗ f) ∘ (μ_{1} ⊗_{t} C)

3. H(μ_{1} ⊗_{t} C) ∘ ρ = ρ ∘ (μ ⊗ H(C))

4. H(Zπ^{a} ⊗ f) ∘ ρ = ρ ∘ (Zπ^{a} ⊗ H(f))

Our main example is the cellular Z-chain complex C = C(F) of the fibre together with a choice C(g) of representatives of the chain homotopy class of chain maps C ⟶ C induced by the fibre transport ω(g). Let α : C^{d−∗} ⟶ C be a representative of the homotopy class of chain maps given by the Poincaré chain equivalence ∩[F] : C^{d−∗} ⟶ C. Recall that the homomorphism v : π ⟶ {±1} sends g to the degree of the automorphism H(ω(g)) of H_{d}(F) ≅ Z. Equip C^{d−∗} with the π-twist C^{d−∗}(g) = v(g) · C(g)^{d−∗}. Then α is a map of chain complexes with a twist.

Recall that Zπ^{∗} = hom_{Zπ}(Zπ, Zπ) carries the Zπ-structure with respect to the wtwisted involution ¯ : Zπ ⟶ Zπ. We shall use the following Zπ-isomorphism as an identification

9.4 ev : Zπ^{∗} ⟶ Zπ f ↦ \overline{f(1)}

Recall that (Zπ ⊗ C)^{d−∗} is equipped with the Zπ-structure with respect to the vw-twisted involution ¯ : Zπ ⟶ Zπ. We shall use the following Zπ-isomorphism as an identification

9.5 Zπ ⊗ C^{d−∗} ⟶ (Zπ ⊗ C)^{d−∗} sending u ⊗ f to v ⊗ x ↦ vf(x)\overline{u}. Given a Zπ-map μ : Zπ ⟶ Zπ, let \overline{μ} : Zπ ⟶ Zπ be the Zπ-map satisfying \overline{μ}(1) = \overline{μ(1)} if ¯ is the w-twisted involution. We have \overline{μ} = μ^{∗} under the identification 9.4. This extends obviously to Zπ-maps μ : Zπ^{a} ⟶ Zπ^{b}. One easily checks that the following data define a symmetric representation (Zπ ⊗ C, Zπ ⊗ α, U) of Zπ with the w-twisted involution into D_{d}(Zπ) with the vw-twisted involution:

9.6 Zπ ⊗ α : Zπ ⊗ C^{d−∗} ⟶ Zπ ⊗ C U : Zπ ⟶ [Zπ ⊗ C, Zπ ⊗ C]_{Zπ}^{op} g ↦ [r(g) ⊗_{t} C]

where r(g) : Zπ ⟶ Zπ maps u to ug. One derives directly from lemma 1.4 and the definitions that the homomorphisms p_{∗} ∘ p^{∗} and (Zπ ⊗ C, Zπ ⊗ α, U)^{∗} from L_{n}(Zπ, w) to L_{n+d}(Zπ, vw) agree.

The strategy of the proof of the up-down formula is described as follows. Given λ ∈ L_{n}(Zπ, w), choose an appropriate (−1)^{m}-quadratic Poincaré Zπ-chain complex of dimension d resp. d + 1 representing p_{∗} ∘ p^{∗}(λ) ∈ L_{d+n}(Zπ, w) if n = 2m resp n = 2m + 1. Do algebraic surgery on this chain complex and show that the result is homotopy equivalent to a (−1)^{m}-quadratic Poincaré Zπ-chain complex which represents σ^{∗}(F, ω) ⊗ λ. As algebraic surgery and homotopy equivalence do not change the class of a (−1)^{m}-quadratic Poincaré Zπ-chain complex in the L-group, the up-down formula follows. To do the algebraic surgery and find the right homotopy equivalence, we need some preliminaries. Namely, we are going to construct the following data if d = 2k respectively d = 2k + 1:

9.7 1. A finitely generated free Z-chain complex D of dimension k. If d is 2k, the differential d_{k} is injective.

2. A chain epimorphism p : C ⟶ D which is k-connected. If d is 2k + 1, then H_{k}(p) has tors H_{k}(C) as kernel.

3. Chain maps D(g) : D ⟶ D for g ∈ π such that p ∘ C(g) and D(g) ∘ p agree. Let cone(p)(g) : cone(p) ⟶ cone(p) be the chain map given by C(g)_{∗−1} ⊕ D(g)_{∗}. Then {D(g)} and {cone(p)(g)} define π-twists.

4. A finitely generated free chain complex E. If d is 2k, then E is concentrated in dimension k and E_{k} is H_{k}(C)/tors H_{k}(C). If d is 2k + 1, then E is concentrated in dimensions k + 1 and k, the differential e_{k+1} is injective and E is a resolution of tors H_{k}(C).

5. A chain map q : C ⟶ E such that H_{k}(q) : H_{k}(C) ⟶ H_{k}(E) = H_{k}(C)/tors H_{k}(C) is the canonical projection if d = 2k and H_{k}(q) : H_{k}(C) ⟶ H_{k}(E) = tors H_{k}(C) induces the identity on tors H_{k}(C) if d = 2k + 1.

6. A π-twist E(g) on E. Let η : Σ^{−1}cone(p) ⟶ E be the composition of q : C ⟶ D with the canonical projection Σ^{−1}cone(p) ⟶ C. Then η is a map of chain complexes with π-twist.

7. We get chain maps ξ : D^{d−∗} ⟶ Σ^{−1}cone(p) by (αp^{d−∗}, 0) and ν : cone(ξ) ⟶ E by (0, η)^{tr}.

8. There is for all i a short exact sequence

0 ⟶ H_{i}(D^{d−∗}) \underrightarrow{H_{i}(ξ)} H_{i}(Σ^{−1}cone(p)) \underrightarrow{H_{i}(η)} H_{i}(E) ⟶ 0 .

This implies that ν is a homotopy equivalence.

Roughly speaking, the data about D and p will be used for the algebraic surgery and the data about E and q for the homotopy equivalence of quadratic Poincaré complexes. We construct this data in the case d = 2k first.

Let D be
… ⟶ {0} ⟶ im(c_{k}) ↪ C_{k−1} ⟶ C_{k−2} ⟶ … ⟶ C_{0}

Define p : C ⟶ D by p_{i} = 0 for i > k, p_{k} = c_{k} and p_{i} = id for i < k. For g, h ∈ π choose a homotopy H(g, h) : C(g) ∘ C(h) ≃ C(gh). There are maps of graded modules D(g) : D_{∗} ⟶ D_{∗} of degree zero and K(g, h) : D_{∗} ⟶ D_{∗+1} of degree with the property that p ∘ C(g) = D(g) ∘ p and K(g, h) ∘ p = p ∘ H(g, h) hold. As p is surjective, D(g) is a chain map and K(g, h) a homotopy between D(g) ∘ D(h) and D(gh). Obviously D(1) and C(1) are the identity. Now one easily checks that items 1.), 2.) and 3.) in the list 9.7 are satisfied.

Define E as required in item 4.) of 9.7. Since im(c_{k}) ⊂ C_{k−1} is free and the sequence 0 ⟶ ker(c_{k}) ⟶ C_{k} ⟶ im(c_{k}) ⟶ 0 is exact, there is a Z-map r : C_{k} ⟶ ker(c_{k}) whose restriction to ker(c_{k}) is the identity. Let pr : ker(c_{k}) ⟶ H_{k}(C)/tors H_{k}(C) be the canonical projection. Define a chain map q : C ⟶ E by q_{k} = pr ∘ r. Obviously item 5.) in 9.7 holds. Define the π-twist on E by the Zπ structure on E_{k} = H_{k}(C)/tors H_{k}(C). Then q : C ⟶ E is a map of chain complexes with π-twist, as it induces a Zπ-map on homology. This proves item 6.) of 9.7.

The compositions p ∘ α ∘ p^{d−∗} is zero because of item 1.) in 9.7 since D_{i}^{d−∗} = {0} for i < k, D_{i} = {0} for i > k and d_{k} is injective. Hence ξ : D^{d−∗} ⟶ cone(p) given by (α ∘ p^{d−∗}, 0)^{tr} is a chain map. Items 1.) and 2.) of 9.7 imply that H_{i}(Σ^{−1}cone(p)) is {0} for i ≤ k − 1 and the canonical map H_{k}(cone(p)) ⟶ H_{k}(C) is injective with tors H_{k}(C) as image. Therefore H_{k}(η ∘ ξ) is zero. Since E is concentrated in one dimension η ∘ ξ is zero and ν : cone(ξ) ⟶ E given by (0, η)^{tr} is a chain map. This proves item 7.)

Next we prove the exactness of the sequence in item 8.). Note that this together with the long homology sequence of ξ implies that ν is a homology equivalence, and hence a homotopy equivalence. Since Σ^{−1}cone(p) is (k−1)-connected, we can choose a chain complex X homotopy equivalent to Σ^{−1}cone(p) such that X_{i} = {0} for i ≤ k − 1. The mapping cone of p^{d−∗} : D^{d−∗} ⟶ C^{d−∗} is (Σ^{−1}cone((−1)^{d+1} · p))^{d−∗} ≃ X^{d−∗}. Obviously H_{i}(X^{d−∗}) = {0} for i ≥ k + 1 and H_{k}(X^{d−∗}) ⊂ (X^{d−∗})_{k} is free. We derive from the long homology sequence of p^{d−∗} that H_{i}(p^{d−∗}) is bijective for i ≥ k + 1 and the following sequence is exact

0 ⟶ H_{k}(D^{d−∗}) \underrightarrow{H_{k}(p^{d−∗})} H_{k}(C^{d−∗}) ⟶ H_{k}(X^{d−∗}) ⟶ 0

Since H_{k}(D^{d−∗}) ⊗ Q is isomorphic to H_{k}(D) ⊗ Q and H_{k}(D) = {0} by item 2.) in 9.7, the module H_{k}(D^{d−∗}) is torsion. This implies that H_{k}(p^{d−∗}) : H_{k}(D^{d−∗}) ⟶ H_{k}(C^{d−∗}) is injective and has tors H_{k}(C^{d−∗}) as image. Since H_{i}(α) : H_{i}(C^{d−∗}) ⟶ H_{i}(C) is bijective, H_{i}(ξ) : H_{i}(D^{d−∗}) ⟶ H_{i}(cone(p)) is bijective for i ≥ k. Now item 8.) in 9.7 follows. This finishes the verification and construction of the data 9.7 in the even-dimensional case d = 2k.

Next we treat the case d = 2k + 1. Let K be the kernel of the canonical projection ker(c_{k}) ⟶ H_{k}(C)/tors H_{k}(C). As its image is free K is a direct summand in ker(c_{k}) and hence in C_{k}. In particular K and C_{k}/K are free and we can choose a retraction r : C_{k} ⟶ K. Let D be

… ⟶ {0} ⟶ C_{k}/K ⟶^{c_{k}} C_{k−1} ⟶ C_{k−2} ⟶ … ⟶ C_{0}

and p : C ⟶ D be the obvious projection. One verifies items 1.) 2.) and 3.) of 9.7 as done in the case d = 2k above.

Choose E as required in item 4.) of 9.7. Let ϵ : E_{k} ⟶ H_{k}(E) = tors H_{k}(C) be the canonical projection. Let pr : K ⟶ tors H_{k}(C) be the canonical epimorphism. Choose \overline{pr} : K ⟶ E_{k} satisfying ϵ ∘ \overline{pr} = pr. Put q_{k} : C_{k} ⟶ E_{k} to be \overline{pr} ∘ r where r : C_{k} ⟶ K is a retraction. Because ϵ ∘ q_{k} ∘ c_{k+1} vanishes, we can choose q_{k+1} : C_{k+1} ⟶ E_{k+1} such that e_{k+1} ∘ q_{k+1} = q_{k} ∘ c_{k+1} holds. Hence we obtain a chain map q : C ⟶ E satisfying item 5.) of 9.7.

Since E is a resolution of the Zπ-module tors H_{k}(C), we may choose chain maps E(g) : E ⟶ E inducing multiplication with g on homology for each g ∈ π. As Σ^{−1}cone(p) is (k − 1)-connected and H_{k}(η) : H_{k}(Σ^{−1}cone(p)) ⟶ H_{k}(E) is a Zπ-homomorphism, η is a map of chain complexes with a π-twist. This proves item 6.) of 9.7.

The compositions p ∘ α ∘ p^{d−∗} and q ∘ α ∘ p^{d−∗} are zero for dimension reasons. Hence item 7.) of 9.7 is true.

It remains to verify item 8.). Because of item 1.) and 2.) of 9.7 it suffices to show that H_{i}(p^{d−∗}) : H_{i}(D^{d−∗}) ⟶ H_{i}(C^{d−∗}) is bijective for i ≥ k + 1. The mapping cone of p^{d−∗} is (Σ^{−1}cone((−1)^{d+1} · p))^{d−∗}). Since H_{i}(Σ^{−1}cone(p)) is zero for i ≤ k − 1 and is torsion for i = k, the universal coefficient theorem shows that H_{i}(Σ^{−1}cone(p)^{d−∗}) = {0} for i ≥ k + 1. One may derive this also by an argument as in the case d = 2k above using a chain complex X which is homotopy equivalent to Σ^{−1}cone(p) and satisfies X_{i} = {0} for i ≤ k − 1. This finishes the construction and verification of the data 9.7.

Now we are ready to prove the up-down formula 2.7 in even base dimensions n = 2m. Represent λ ∈ L_{n}(Zπ, w) by a nonsingular (−1)^{m}-quadratic form μ : Zπ^{a} ⟶ Zπ^{a} over Zπ with respect to the w-twisted involution, i.e. a Zπ-map μ such that μ + (−1)^{m}\overline{μ} is an isomorphism. We obtain a (−1)^{m}-quadratic structure {ψ_{s}} on Zπ^{a} ⊗ C if we define ψ_{0} = (μ ⊗_{t} C) ∘ (Zπ^{a} ⊗ α) : Zπ^{a} ⊗ C^{d−∗} ⟶ Zπ^{a} ⊗ C and ψ_{s} = 0 for s ≥ 1. The class of the d-dimensional (−1)^{m}-quadratic Poincaré Zπ-chain complex (Zπ^{a} ⊗ C, {ψ_{s}}) in L_{d+n}(Zπ, vw) is p_{∗} ∘ p^{∗}(λ).

The composition (Zπ^{a} ⊗ p) ∘ (ψ + (−1)^{m} · ψ^{d−∗}) ∘ (Zπ^{a} ⊗ p^{d−∗}) is zero by item 1.) of 9.7. Hence {ψ_{s}} ∈ Q_{d}(Zπ^{a} ⊗ C) can be extended to {(∂ψ, ψ)_{s}} ∈ Q_{d+1}(Zπ^{a} ⊗ p) by ∂ψ = 0. Denote by (\overline{C}, \{\overline{ψ}\}) the result under surgery on (Zπ^{a} ⊗ p, \{(∂ψ, ψ)_{s}\}) as defined in Ranicki [30], section 1.5. We obtain a chain map

ξ̃: Zπ^{a} ⊗ D^{d−∗} ⟶ Σ^{−1}cone(Zπ^{a} ⊗ p) = Zπ^{a} ⊗ Σ^{−1}cone(p)

by ((ψ_{0} + (−1)^{m} · ψ_{0}^{d−∗}) ∘ (Zπ^{a} ⊗ p^{d−∗}), 0)^{tr}. By definition \overline{C} is the mapping cone of ξ̃. Let

η̃ : Zπ^{a} ⊗ Σ^{−1}cone(p) ⟶ Zπ^{a} ⊗ E

be Zπ^{a} ⊗ η. One computes H_{k}(η̃ ∘ ξ̃) = Zπ^{a} ⊗ H_{k}(η ∘ ξ). using the identification 9.2 and lemma 9.3. Since H_{k}(η ∘ ξ) vanishes by item 8.) of 9.7 and E is concentrated in one dimension by item 4.) of 9.7, η̃ ∘ ξ̃ is zero. Hence we obtain a chain map

ν̃ : \overline{C} ⟶ Zπ^{a} ⊗ E

by (0, η̃). If we choose a Z-base for C and D, we get an induced Zπ-base for the source and target of ν̃ and the Whitehead torsion τ(ν̃) ∈ K̃_{1}(Zπ) of ν̃ is defined. It is independent of the choice of the Z-base above, since we work in the reduced K_{1}-group. Recall from Lück [15], section 5 that associated to the chain complex with π-twist (D^{d−∗}, {D^{d−∗}(g)}) there is a K-theory transfer (D^{d−∗}, {D^{d−∗}(g)})^{∗} : K̃_{1}(Zπ) ⟶ K̃_{1}(Zπ) (and also for K_{0}(Zπ)).

Lemma 9.8 ν̃ : \overline{C} ⟶ Zπ^{a} ⊗ E is a homotopy equivalence. Its Whitehead torsion is the image under (D^{d−∗}, \{D^{d−∗}(g)\})^{∗} of the class in K̃_{1}(Zπ) represented by μ + (−1)^{m} · \overline{μ}.

\underline{\underline{Proof}} : There is a homotopy H from ξ̃ to (Zπ^{a} ⊗ ξ) ∘ (μ ⊗_{t} D^{d−∗} + (−1)^{m} · \overline{μ} ⊗_{t} D^{d−∗}) because of lemma 9.3. We obtain a chain map

ν′ = ( \begin{matrix}μ ⊗_{t} D^{d−∗} + (−1)^{m} · \overline{μ} ⊗_{t} D^{d−∗} & 0 \\ H & id \end{matrix} ) : \overline{C} = cone(ξ̃) ⟶ cone(Zπ^{a} ⊗ ξ)

As μ ⊗_{t} D^{d−∗} + (−1)^{m} · \overline{μ} ⊗_{t} D^{d−∗} is a homotopy equivalence, ν′ is a homotopy equivalence. The Whitehead torsion of ν′ is the Whitehead torsion of μ ⊗_{t} D^{d−∗} + (−1)^{m} · \overline{μ} ⊗_{t} D^{d−∗} which is by definition the image under (D^{d−∗}, {D^{d−∗}(g)})^{∗} of the class in K̃_{1}(Zπ) represented by μ + (−1)^{m} · \overline{μ}. One can identify the target of ν′ with Zπ^{a} ⊗ cone(ξ). As ν is by item 8.) of 9.7 a homotopy equivalence Zπ^{a} ⊗ ν : Zπ^{a} ⊗ cone(ξ) ⟶ Zπ^{a} ⊗ E is a simple homotopy equivalence. Its composition with ν′ is ν̃ and the claim follows.

Let {ϕ_{s}} be the quadratic structure on Zπ^{a} ⊗E obtained by pulling back the quadratic structure \{\overline{ψ}_{s}\} by ν̃, i.e., ϕ_{s} = ν̃ ∘ \overline{ψ}_{s} ∘ ν̃^{d−∗}. As ν̃ is (0, q, 0) : D^{d−∗−1} ⊕ C_{∗} ⊕ D_{∗+1} ⟶ E_{∗} one easily checks from the definition of \overline{ψ} in Ranicki [30], section 1.5 that ϕ_{s} is zero for s ≥ 1 and ϕ_{0} is q ∘ (μ ⊗_{t} C) ∘ (Zπ^{a} ⊗ α) ∘ Zπ^{a} ⊗ q^{d−∗}. Hence

H_{i}(ϕ_{0}) : H_{i}(Zπ^{a} ⊗ E^{d−∗}) ⟶ H_{i}(Zπ^{a} ⊗ E)

can be identified using 9.2, lemma 9.3 and item 4.) of 9.7 with

μ ⊗ ϕ_{F} : Zπ^{a} ⊗ hom_{Z}(H_{k}(F)/tors H_{k}(F), Z) ⟶ Zπ^{a} ⊗ H_{k}(F)/tors H_{k}(F)

if d = 2k and ϕ_{F} is the intersection form on F and with

μ ⊗ ϕ_{F} : Zπ^{a} ⊗ hom_{Z}(tors H_{k}(F), Q/Z) ⟶ Zπ^{a} ⊗ tors H_{k}(F)

if d = 2k + 1 and ϕ_{F} is the linking form. The class of the d-dimensional (−1)^{m}-quadratic Poincaré Zπ-chain complex (Zπ^{a} ⊗ E, {ϕ_{s}}) in L_{n+d}(Zπ, vw) = L_{d}(Zπ, (−1)^{m} · vw) has been shown to be σ^{∗}(F, ω) ⊗ λ. This finishes the proof of the up-down formula 2.7 in the case of even base dimension n = 2m for L^{h}.

Finally, we make some remarks how this extends to the intermediate L-groups in case of K_{1}-decorations. The case K_{0} is completely analogous. Given a finitely generated free Z-chain complex with a π-twist D, {D(g)}, let s(D) ∈ Sw(π, Z) be the class ∑_{i≥0} (−1)^{i} · [H_{i}(D)]. The assumptions 7.1 just say that s(D) and s(C) lie in Z ⊂ Sw(π, Z) where C and D are the chain complexes with a π-twist defined in 9.7. The long homology sequence of ξ becomes an exact Zπ-sequence if one identifies H_{i}(cone(ξ)) with H_{i}(E) by H_{i}(ν). The long exact sequence of p is also compatible with the Zπ-structures. This implies s(E) = s(C) − s(D) + s(D^{d−∗}). By the universal coefficient theorem (−1)^{d} · s(D^{d−∗}) is the image of s(D) under the v-twisted involution. Thus s(D^{d−∗}) and [tors H_{k}(F)] = (−1)^{k} · s(E) lie in Z ⊂ Sw(π, Z) and σ^{∗}(F, ω) defines an element in L_{Z}^{d}(π, Z, v). The K-theoretic transfer homomorphism (D^{d−∗}, {D^{d−∗}(g)})^{∗} : K̃_{1}(Zπ) ⟶ K̃_{1}(Zπ) sends X to Y′ since it is given by the pairing 2.1 and s(D) ∈ Sw(π, Z). Now one easily checks that the proof above for the up-down formula for L^{h} goes through for the intermediate L-groups.

10. Proof of the up-down Formula in Odd Base Dimensions

Next, we finish the proof of the up-down formula 2.7 in the case where the base dimension is odd, n = 2m + 1. We give two different proofs. The first one is based on the Shaneson splitting and is comparatively short but does not carry over to the intermediate L-groups. The second one is a blown up version of the proof in the even-dimensional case and holds also for the intermediate L-groups. We restrict ourselves to the proof for L^{h}.

Let i : L_{n}^{h}(Zπ, w) ⟶ L_{n+1}^{s}(Z[Z × π], w) be the homomorphism given by the tensorproduct with the symmetric Z[Z]-chain complex of the universal covering of S^{1}. Since the Euler characteristic of S^{1} is zero the image lies in the L^{s}-groups. This map is a split injection by the results of Shaneson [33] and Ranicki [28]. Let (Zπ ⊗ C, Zπ ⊗ α, U) be the symmetric representation 9.6 whose surgery transfer (Zπ⊗C, Zπ⊗α, U)^{∗} is just p_{∗} ∘ p^{∗}. The Whitehead torsion of Zπ^{a} ⊗ α : Zπ^{a} ⊗ C^{d−∗} ⟶ Zπ^{a} ⊗ C is zero. Hence (Zπ ⊗ C, Zπ ⊗ α, U) induces transfer maps on both L^{h} and L^{s}. One easily verifies that the following square commutes

L_{n}^{h}(Zπ, w) \underrightarrow{i} L_{n+1}^{s}(Z[π × Z], w)
(Zπ ⊗ C, Zπ ⊗ α, U)_{∗} ↓  ↓ (Zπ ⊗ C, Zπ ⊗ α, U)_{∗}
L_{n+d}^{h}(Zπ, vw) \underrightarrow{i} L_{n+d+1}^{s}(Z[π × Z], vw)

One also checks easily that this square remains comutative if we substitute the vertical arrows by σ^{∗}(F, ω)⊗?. The up-down formula for L^{h} in the odd-dimensional case n = 2m + 1 now follows from the up-down formula for L^{s} in the even case n = 2m + 2 and injectivity of i. A similar argument applies to L^{p} instead of L^{h}.

Next we give a different proof which is much more complicated but carries over to the intermediate L-groups as well. Recall that a nonsingular (−1)^{m}-quadratic formation (Zπ^{a}, Zπ^{b}, μ, γ) consists of Zπ-homomorphisms μ : Zπ^{a} ⟶ Zπ^{b} and γ : Zπ^{b} ⟶ Zπ^{a} such that there is a Zπ-map θ : Zπ^{b} ⟶ Zπ^{b} with the properties that θ − (−1)^{m} · \overline{θ} = μ ∘ γ holds and the following square is cartesian:

10.1

Zπ^{b} \underrightarrow{γ} Zπ^{a}
−\overline{μ} ↓  _{↓ μ}
Zπ^{a} \underrightarrow{\overline{γ}} Zπ^{b}

Consider an element λ ∈ L_{2m+1}(Zπ, vw). Represent λ by a nonsingular (−1)^{m}-quadratic formation (Zπ^{a}, Zπ^{b}, μ, γ). Choose a Zπ-chain map θ : D^{d−∗} ⟶ D and a Zπ-homotopy χ from θ ⊗_{t} C − (−1)^{m} · \overline{θ} ⊗_{t} C to (μ ⊗_{t} C) ∘ (γ ⊗_{t} C) ∘ (Zπ^{b} ⊗ α). Equip cone(μ ⊗_{t} C) with the (−1)^{m}-quadratic structure {ψ_{s}} defined accordingly to 8.4 as follows

10.2

ψ_{0} = ( \begin{matrix}0 & γ ⊗_{t} C ∘ Zπ^{b} ⊗ α \\ 0 & χ \end{matrix} ) : cone(μ ⊗_{t} C)^{d+1−∗} ⟶ cone(μ ⊗_{t} C)_{∗}

ψ_{1} = ( \begin{matrix}0 & 0 \\ 0 & θ ⊗_{t} C ∘ Zπ^{b} ⊗ α \end{matrix} ) : cone(μ ⊗_{t} C)^{d+1−∗} ⟶ cone(μ ⊗_{t} C)_{∗−1}

ψ_{s} = 0 for s ≥ 2

Then the class of the (−1)^{m}-quadratic Poincaré Zπ-chain complex (cone(μ ⊗_{t} C), {ψ_{s}}) of dimension d + 1 in L_{n+d}(Zπ, vw) = L_{d+1}(S, (−1)^{m} · vw) is p_{∗} ∘ p^{∗}(λ).

The following square commutes (strictly, not only up to homotopy) by item 3.) in 9.7

10.3

Zπ^{a} ⊗ C \underrightarrow{Zπ^{a} ⊗ p} Zπ^{a} ⊗ D
μ ⊗_{t} C ↓  _{↓ μ ⊗_{t} D}

Zπ^{b} ⊗ C \underrightarrow{Zπ^{b} ⊗ p} Zπ^{b} ⊗ D

We get a chain map p̂: cone(μ ⊗_{t} C) ⟶ cone(μ ⊗_{t} D) by

( \begin{matrix}Zπ^{a} ⊗ p_{∗−1} & 0 \\ 0 & Zπ^{b} ⊗ p_{∗} \end{matrix} ) : Zπ^{a} ⊗ C_{∗−1} ⊕ Zπ^{b} ⊗ C_{∗} ⟶ Zπ^{a} ⊗ D_{∗−1} ⊕ Zπ^{b} ⊗ D_{∗}

By item 1.) in 9.7 p̂ ∘ ψ_{s} ∘ p̂^{d−∗} is zero for all s ≥ 0. Hence {ψ_{s}} ∈ Q_{d+1}(cone(μ ⊗_{t} C)) extends to \{(∂ψ, ψ)_{s}\} ∈ Q_{d+2}(p̂) if we put ∂ψ to be zero. Let (\overline{C}, \{\overline{ψ_{s}}\}) be the result of algebraic surgery on (p̂, {(∂ψ, ψ)_{s}}). Because p̂ ∘ (ψ_{0} + (−1)^{m}ψ_{0}^{d−∗}) ∘ p̂^{d+1−∗} is zero by item 1.) in 9.7, we obtain a chain map

ξ̃: cone(\overline{μ} ⊗_{t} D^{d−∗}) ⟶ Σ^{−1}cone(p̂)

by ((ψ_{0} + (−1)^{m}ψ_{0}^{d−∗}) ∘ p̂^{d+1−∗}, 0)^{tr}. By definition cone(ξ̃) is \overline{C}. The following diagram commutes up to homotopy by item 6.) in 9.7 and lemma 9.3.

Zπ^{a} ⊗ Σ^{−1}cone(p) \underrightarrow{Zπ^{a} ⊗ η} Zπ^{a} ⊗ E
μ ⊗_{t} Σ^{−1}cone(p) ↓  _{↓ μ ⊗_{t} E}

Zπ^{b} ⊗ Σ^{−1}cone(p) \underrightarrow{Zπ^{b} ⊗ η} Zπ^{b} ⊗ E

Choose a corresponding homotopy H. We get a Zπ-chain map

η̃ = ( \begin{matrix}Zπ^{a} ⊗ η & 0 \\ H & Zπ^{b} ⊗ η \end{matrix} ) : Σ^{−1}cone(p̂) = cone(μ ⊗_{t} Σ^{−1}cone(p)) ⟶ cone(μ ⊗_{t} E)

We derive from item 1.) of 9.7 that η̃ ∘ ξ̃ is zero. Hence we obtain a chain map

ν̃ : \overline{C} = cone(ξ̃) ⟶ cone(μ ⊗_{t} E)

by (η̃, 0)^{tr}.

Lemma 10.4 ν̃ : \overline{C} ⟶ cone(μ ⊗_{t} E) is a homotopy equivalence. Its Whitehead torsion is the image under the K-theory transfer (D^{d−∗}, {D^{d−∗}(g)})^{∗} of the Whitehead torsion of the nonsingular (−1)^{m}-quadratic formation (Zπ^{a}, Zπ^{b}, μ, γ).

\underline{\underline{Proof}} : We use the following abbreviations :

f_{1} = (γ ⊗_{t} C) ∘ (Zπ^{b} ⊗ α) ∘ (Zπ^{b} ⊗ p^{d−∗}) : Zπ^{b} ⊗ D^{d−∗} ⟶ Zπ^{a} ⊗ C f_{2} = (−1)^{m} · (Zπ^{b} ⊗ α^{d−∗}) ∘ (\overline{γ} ⊗_{t} C^{d−∗}) ∘ (Zπ^{a} ⊗ p^{d−∗}) : Zπ^{a} ⊗ D^{d−∗} ⟶ Zπ^{b} ⊗ C

g_{1} = (Zπ^{a} ⊗ α) ∘ (Zπ^{a} ⊗ p^{d−∗}) : Zπ^{a} ⊗ D^{d−∗} ⟶ Zπ^{a} ⊗ C g_{2} = (−1)^{m} · (Zπ^{b} ⊗ α) ∘ (Zπ^{b} ⊗ p^{d−∗}) : Zπ^{b} ⊗ D^{d−∗} ⟶ Zπ^{b} ⊗ C Then ξ̃: cone(\overline{μ} ⊗_{t} D^{d−∗}) ⟶ Σ^{−1}cone(p̂) is given by

( \begin{matrix}f_{1} & 0 \\ ∗ & f_{2} \\ 0 & 0 \end{matrix} ) : (Zπ^{b}⊗D^{d−(∗−1)})⊕(Zπ^{b}⊗D^{d−∗}) → (Zπ^{a}⊗C_{∗−1})⊕(Zπ^{b}⊗C_{∗})⊕Σ^{−1}cone(μ⊗_{t}D)_{∗}

As the square 10.1 is cartesian, we can find a cartesian square

10.5  \begin{matrix}Zπ^{a} & \underrightarrow{δ} & Zπ^{b} \\ μ ↓ & & ↓ −\overline{μ} \\ Zπ^{b} & \underrightarrow{ϵ} & Zπ^{a} \end{matrix}

and a Zπ-map Ω : Zπ^{b} ⟶ Zπ^{a} satisfying

10.6 μ ∘ Ω = id − \overline{γ} ∘ ϵ Ω ∘ μ = id − γ ∘ δ

If one regards the square 10.1 as a chain homotopy equivalence from the first to the second column, the square 10.5 describes a chain homotopy inverse and Ω a chain homotopy. Define a chain map κ : cone(μ ⊗_{t} D^{d−∗}) ⟶ cone(\overline{μ} ⊗_{t} D^{d−∗}) by

( \begin{matrix}δ ⊗_{t} D^{d−(∗−1)} & 0 \\ ∗ & ϵ ⊗_{t} D^{d−∗} \end{matrix} ) : (Zπ^{a} ⊗ D^{d−(∗−1)}) ⊕ (Zπ^{b} ⊗ D^{d−∗}) ⟶

(Zπ^{b} ⊗ D^{d−(∗−1)}) ⊕ (Zπ^{a} ⊗ D^{d−∗})

where ∗ denotes any homotopy from (\overline{μ} ⊗_{t} D^{d−∗}) ∘ (δ ⊗_{t} D^{d−∗}) to (ϵ ⊗_{t} D^{d−∗}) ∘ (μ ⊗_{t} D^{d−∗}). Such a homotopy ∗ exists as the square 10.5 is commutative. Because of the relations 10.6 and lemma 9.3 we can choose homotopies

h_{1} : f_{1} ∘ (δ ⊗_{t} D^{d−∗}) + g_{1} ∘ (Ω ⊗_{t} D^{d−∗}) ∘ (μ ⊗_{t} D^{d−∗}) ≃ g_{1} h_{2} : f_{2} ∘ (ϵ ⊗_{t} D^{d−∗}) + g_{2} ∘ (μ ⊗_{t} D^{d−∗}) ∘ (Ω ⊗_{t} D^{d−∗}) ≃ g_{2}

Define a map of degree one H : cone(μ ⊗_{t} D^{d−∗}) ⟶ Σ^{−1}cone(p̂) by

( \begin{matrix}h_{1} & g_{1} ∘ (Ω ⊗_{t} D^{d−∗}) \\ 0 & −h_{2} \\ 0 & 0 \end{matrix} ) : (Zπ^{a} ⊗ D^{d−(∗−1)}) ⊕ (Zπ^{b} ⊗ D^{d−∗}) ⟶

(Zπ^{a} ⊗ C_{∗−1}) ⊕ (Zπ^{b} ⊗ C_{∗}) ⊕ (Σ^{−1}cone(μ ⊗_{t} C)_{∗})

One checks using item 1.) and 2.) of 9.7 that the compositions of Zπ^{a} ⊗ p and Zπ^{a} ⊗ q with h_{1} and H as well as the compositions of Zπ^{b} ⊗ p and Zπ^{b} ⊗ q with h_{2} are zero. Now one easily verifies that H is a homotopy between ξ̃∘ κ̃ and a chain map g̃ : cone(μ ⊗_{t} D^{d−∗}) ⟶ Σ^{−1}cone(p̂) of the following shape

( \begin{matrix}g_{1} & 0 \\ ∗ & g_{2} \\ 0 & 0 \end{matrix} ) : (Zπ^{a} ⊗ D^{d−(∗−1)}) ⊕ (Zπ^{b} ⊗ D^{d−∗})

⟶ (Zπ^{a} ⊗ C_{∗−1}) ⊕ (Zπ^{b} ⊗ C_{∗}) ⊕ (Σ^{−1}cone(μ ⊗_{t} D)_{∗}) .

We have constructed a diagram which commutes up to the homotopy H.

cone(μ ⊗_{t} D^{d−∗}) \underrightarrow{g̃} Σ^{−1}cone(p̂)
κ̃ ↓  ↓ id
cone(\overline{μ} ⊗_{t} D^{d−∗}) \underrightarrow{ξ̃} Σ^{−1}cone(p̂)

Let ν′ : cone(g̃) ⟶ cone(ξ̃) = \overline{C} be the induced chain map (see 8.5). It is a homotopy equivalence as the square 10.5 is cartesian and therefore κ̃ is a homotopy equivalence. The Whitehead torsion of ν′ is the Whitehead torsion of κ̃ which can be identified with the negative of the image of the Whitehead torsion of the cartesian square 10.1 under the K-theory transfer (D^{d−∗}, {D^{d−∗}(g)})^{∗}. The mapping cone of g̃ can be identified with the mapping cone of the following chain map ĝ : Zπ^{a} ⊗ cone(ξ) ⟶ Zπ^{b} ⊗ cone(ξ) given by

( \begin{matrix}μ ⊗_{t} D^{d−∗} & 0 & 0 \\ ∗ & μ ⊗_{t} C & 0 \\ 0 & 0 & μ ⊗_{t} D \end{matrix} ) : (Zπ^{a} ⊗ D^{d−(∗−1)}) ⊕ (Zπ^{a} ⊗ C_{∗}) ⊕ (Zπ^{a} ⊗ D_{∗+1})

⟶ (Zπ^{b} ⊗ D^{d−(∗−1)}) ⊕ (Zπ^{b} ⊗ C_{∗}) ⊕ Zπ^{b} ⊗ D_{∗+1}

The composition of ν̃ with ν′ is given by the following up to homotopy commutative square together with an appropiate choice of homotopy (see 8.5).

Zπ^{a} ⊗ cone(ξ) \underrightarrow{Zπ^{a} ⊗ ν} Zπ^{a} ⊗ E
ĝ ↓  _{↓ μ ⊗_{t} E}

Zπ^{b} ⊗ cone(ξ) \underrightarrow{Zπ^{b} ⊗ ν} Zπ^{b} ⊗ E

Since ν is a homotopy equivalence by item 8.) of 9.7 Zπ^{a} ⊗ ν and Zπ^{b} ⊗ ν and hence ν̃ ∘ ν′ are Zπ-homotopy equivalences with trivial Whitehead torsion. This finishes the proof of lemma 10.4.

Equip cone(μ ⊗_{t} E) with the quadratic structure \{ϕ_{s}\} = \{ν ∘ \overline{ψ}_{s} ∘ ν^{d+1−∗}\}. One computes directly from 10.2 that ϕ is given by :

ϕ_{0} = ( \begin{matrix}0 & (Zπ^{a} ⊗ q) ∘ (γ ⊗_{t} C) ∘ (Zπ^{b} ⊗ α) ∘ (Zπ^{b} ⊗ q^{d−∗}) \\ 0 & (Zπ^{b} ⊗ q) ∘ χ ∘ (b ⊗_{t} q^{d−∗}) \end{matrix} )

ϕ_{1} = ( \begin{matrix}0 & 0 \\ 0 & (Zπ^{b} ⊗ q) ∘ (Zπ^{b} ⊗ θ) ∘ (Zπ^{b} ⊗ α) ∘ (Zπ^{b} ⊗ q^{d−∗}) \end{matrix} )

ϕ_{s} = 0 for s ≥ 2

We can identify

H_{k}((Zπ^{a} ⊗ q) ∘ (γ ⊗_{t} C) ∘ (Zπ^{b} ⊗ α) ∘ (Zπ^{b} ⊗ p^{d−∗})) : H_{k}(Zπ^{b} ⊗ E) ⟶ H_{k}(Zπ^{a} ⊗ E)

using the identification 9.2, lemma 9.3 and and item 4.) of 9.7 with

γ ⊗ ϕ_{F} : Zπ^{b} ⊗ hom_{Z}(H_{k}(F)/tors H_{k}(F), Z) ⟶ Zπ^{a} ⊗ H_{k}(F)/tors H_{k}(F)

if d = 2k and ϕ_{F} is the intersection form on F, and with

γ ⊗ ϕ_{F} : Zπ^{a} ⊗ hom_{Z}(tors H_{k}(F), Q/Z) ⟶ Zπ^{a} ⊗ tors H_{k}(F)

if d = 2k + 1 and ϕ_{F} is the linking form. Analogously we can identify

H_{k}(μ ⊗_{t} E) : H_{k}(Zπ^{a} ⊗ E) ⟶ H_{k}(Zπ^{b} ⊗ E)

with

μ ⊗ id : Zπ^{a} ⊗ hom_{Z}(H_{k}(F)/tors H_{k}(F), Z) ⟶ Zπ^{a} ⊗ H_{k}(F)/tors H_{k}(F)

if d = 2k, and with

μ ⊗ id : Zπ^{a} ⊗ hom_{Z}(tors H_{k}(F), Q/Z) ⟶ Zπ^{a} ⊗ tors H_{k}(F)

if d = 2k + 1. Now it follows from the definitions that (E, {ϕ_{s}}) represents σ^{∗}(F, ω) ⊗ λ in L_{d+n}(Zπ, vw). This finishes the proof of the up-down formula 2.7.

11. The Pairing 2.4 is Well-Defined

This section is devoted to the proof that the pairing 2.4 is well-defined. We use the notation of section 3. Let (M, ψ) be a nonsingular (π, v)-equivariant (−1)^{k+1}-symmetric linking form over Z. We have to show that the transfer (Zπ ⊗_{Z} F_{∗}, id ⊗ α, U)^{∗} is trivial if (M, ψ) is hyperbolic or a boundary. The strategy of the proof is in all cases the following. We represent λ ∈ L_{n}(Zπ, w) by a nonsingular quadratic form or formation and look at a Poincaré complex (C, {ψ_{s}}) representing the image of λ under (Zπ ⊗_{Z} F_{∗}, id ⊗ α, U)^{∗}. We give a chain map p̂: C ⟶ D. Recall that the Q-groups are homology groups of certain chain complexes (see [30], section 1.1). We have the class p̂_{∗}({ψ_{s}}) ∈ Q_{n+d}(D) given by the cycle {p̂ ∘ ψ_{s} ∘ p̂^{n+d−∗}}. We shall specify a chain {∂ψ_{s}} whose image under the differential is just {p̂ ∘ ψ_{s} ∘ p̂^{n+d−∗}}. Hence p̂_{∗}({ψ_{s}}) ∈ Q_{n+d}(D) is zero. This guarantees that we can do surgery on p̂. We leave to the reader the easy verification that the result under surgery is a contractible Poincaré complex. Then the claim follows.

Assume that (M, ψ) is the boundary of (N, ϕ). Choose F to be N ⟶ hom_{Z}(N, Z) and the π-twist to be the Zπ-structure. Moreover, the Poincaré duality map α : F^{1−∗} ⟶ F can be choosen to be the Zπ-chain map which is (−1)^{k} · id in dimension 1 and id in dimension 0. Let 1(N) be the chain complex concentrated in dimension 1 and having N as chain module there. It inherits a π-twist from the Zπ-structure on N. Let p : F ⟶ 1(N) be given by p_{1} = id.

If n = 2m and the nonsingular (−1)^{m}-quadratic form μ : Zπ^{a} ⟶ Zπ^{a} represents λ, we choose p̂ to be Zπ^{a} ⊗ p : C = Zπ^{a} ⊗ F ⟶ D = Zπ^{a} ⊗ 1(N). Then {p̂ ∘ ψ_{s} ∘ p̂^{n+d_{∗}}} is zero for dimension reasons. So we can choose {∂ψ_{s}} to be zero.

If n = 2m + 1 and λ is represented by the nonsingular (−1)^{m}-quadratic formation (Zπ^{a}, Zπ^{b}, μ, γ) we define the chain map p̂: C = cone(μ ⊗_{t} F) ⟶ cone(μ ⊗_{t} 1(N)) by (Zπ^{a} ⊗ p) ⊕ (Zπ^{b} ⊗ p). This is possible, as p is a Zπ-map and so (μ ⊗_{t} 1(N)) ∘ (Zπ^{a} ⊗ p) agrees with (Zπ^{b} ⊗ p) ∘ (μ ⊗_{t} F). Moreover, we can choose the homotopy χ appearing in the definition 8.4 of ψ_{0} of C to be zero . Now one easily checks again that {p̂ ∘ ψ_{s} ∘ p̂^{n+d_{∗}}} is zero.

Now assume that (M, ψ) is hyperbolic. Let

0 ⟶ L \underrightarrow{i} M \underrightarrow{q} hom_{Z}(L, Q/Z) ⟶ 0

be the corresponding sequence. If F_{1} ⟶ F_{0} ⟶^{ϵ} M is the Z-resolution of M, let G_{0} be F_{0}, ϵ′ : G_{0} ⟶ hom_{Z}(L, Q/Z) be the composition q ∘ ϵ and G_{1} be the kernel of ϵ′. There is precisely one chain map p : F ⟶ G satisfying p_{0} = id. If {F (g)} is the π-twist on F, there is precisely one π-twist on G such that G(g) ∘ p agrees with p ∘ F (g) for all g ∈ π.

If n = 2m and the nonsingular (−1)^{m}-quadratic form μ : Zπ^{a} ⟶ Zπ^{a} represents λ, we choose p̂ to be Zπ^{a} ⊗ p : C = Zπ^{a} ⊗ F ⟶ D = Zπ^{a} ⊗ G. Then the chain map {p̂ ∘ ψ_{0} ∘ p̂^{d−∗}} induces (γ ⊗ H(D)) ∘ (Zπ^{a} ⊗ H(p ∘ α ∘ p^{d−∗})) on homology using the identification 9.2 by lemma 9.3. Because G^{d−∗} and G are resolutions and H(p ∘ α ∘ p^{d−∗}) is zero, we can choose a nullhomotopy ∂ψ_{0} for {p̂ ∘ ψ_{0} ∘ p̂^{d−∗}}. Put ∂ψ_{s} to be zero for s ≥ 1.

If n = 2m + 1 and λ is represented by the nonsingular (−1)^{m}-quadratic formation (Zπ^{a}, Zπ^{b}, μ, γ), we define the chain map p̂: C = cone(μ ⊗_{t} F) ⟶ cone(μ ⊗_{t} G) by (Zπ^{a} ⊗ p) ⊕ (Zπ^{b} ⊗ p). Choose nullhomotopies

H_{0} : (Zπ^{a} ⊗ p) ∘ (γ ⊗_{t} F) ∘ (Zπ^{b} ⊗ α) ∘ (Zπ^{b} ⊗ p^{d−∗}) ≃ 0
H_{1} : (Zπ^{b} ⊗ p) ∘ (θ ⊗_{t} F) ∘ (Zπ^{b} ⊗ α) ∘ (Zπ^{b} ⊗ p^{d−∗}) ≃ 0

Let ∂ψ_{s} be given by

∂ψ_{0} = ( \begin{matrix}0 & −H_{0} \\ 0 & 0 \end{matrix} ) : Zπ^{a} ⊗ G^{1−∗} ⊕ Zπ^{b} ⊗ G^{1−(∗−1)} ⟶ Zπ^{a} ⊗ G_{∗} ⊕ Zπ^{b} ⊗ G_{∗+1}

∂ψ_{1} = ( \begin{matrix}0 & 0 \\ 0 & −H_{1} \end{matrix} ) : Zπ^{a} ⊗ G^{1−∗} ⊕ Zπ^{b} ⊗ G^{1−(∗−1)} ⟶ Zπ^{a} ⊗ G_{∗−1} ⊕ Zπ^{b} ⊗ G_{∗}

∂ψ_{s} = 0 for s ≥ 2

If we alter the cycle {p̂ ∘ ψ_{s} ∘ p̂^{n+d−∗}}. by the boundary given by {∂ψ_{s}}, we obtain a new cycle {ψ′_{s}} which represents the same element as {ψ_{s}} in the Q-group and is of the following shape

ψ′_{0} = ( \begin{matrix}0 & 0 \\ 0 & x \end{matrix} ) : Zπ^{a} ⊗ G^{1−∗} ⊕ Zπ^{b} ⊗ G^{1−(∗−1)} ⟶ Zπ^{a} ⊗ G_{∗−1} ⊕ Zπ^{b} ⊗ G_{∗}

ψ′_{s} = 0 for s ≥ 1

Since {ψ′_{s}} is a cycle, x is actually a chain map x : Zπ^{a} ⊗ G^{1−∗} ⟶ Σ^{−1}Zπ^{a} ⊗ G_{∗}. As the first differential of G is injective, x must be zero. Hence {ψ′_{s}} is zero.

Appendix: Characteristic Class Formulae

We shall now use the algebraic L-theory assembly map of Ranicki [32] to relate the expression of 2.7

p_{∗} ∘ p^{∗} = σ^{∗}(F, p) ⊗ − : L_{n}(Z[π_{1}(B)]) ⟶ L_{n+d}(Z[π_{1}(B)])

(where σ^{∗}(F, p) ≡ σ^{∗}(F, ω)) for a fibre bundle F ⟶ E ⟶^{p} B of manifolds with the characteristic class formulae for the signature of E.

In the first instance we recall the results of Atiyah [3], Lusztig [19] and Meyer [20] expressing the twisted signature in terms of characteristic classes, by means of the Atiyah-Singer index theorem.

Atiyah [3] considered a differentiable fibre bundle F ⟶ E ⟶^{p} B of oriented manifolds with dim(F) = 2k. The action of π_{1}(B) on H^{k}(F; R) determines a flat vector bundle Γ over B with the fibres H^{k}(F_{x}; R) (x ∈ B) nonsingular (−)^{k}-symmetric forms over R. For k ≡ 0(mod 2) the bundle splits as Γ = Γ^{+} ⊕ Γ^{−} with the form positive/negative definite on Γ^{±}. For k ≡ 1(mod 2) the Hodge ∗-operator defines a complex structure on Γ, so that there is defined a complex conjugate bundle Γ^{∗}. The topological K-theory signature of Γ is defined by

[Γ]_{K} = \{ \begin{matrix}Γ^{+} − Γ^{−} ∈ KO(B) & if k ≡ 0(mod 2) \\ Γ^{∗} − Γ ∈ KU(B) & if k ≡ 1(mod 2) . \end{matrix}

The twisted signature formula of [3] in the case dim(B) = 2j, dim(E) = 2(j +k) ≡ 0(mod 4) is

sign(E) = ⟨ch([Γ]_{K}) ∪ L̃(B), [B]_{Q}⟩ ∈ Z

with ch the Chern character, [B]_{Q} ∈ H_{2j}(B; Q) the fundamental class and L̃ the modification of the Hirzebruch L-genus defined by

L̃(B) = ∏_{i=1}^{j} \frac{x_{i}}{tanh x_{i}/2} ∈ H^{4∗}(B; Q)

with x_{1}, x_{2}, …, x_{j} notional elements of degree 2 such that the ith Pontrjagin class p_{i}(τ_{B}) ∈ H^{4i}(B; Q) of the tangent bundle τ_{B} is the ith elementary symmetric function in x_{1}^{2}, x_{2}^{2}, …, x_{j}^{2}. The tangent bundle of E is the Whitney sum τ_{E} = i_{∗}τ_{F} ⊕ p^{∗}τ_{B} of the pushforward i_{∗}τ_{F} along the fibre inclusion i : F → E of τ_{F} and the pullback p^{∗}τ_{B} along the projection p of τ_{B}, with

p_{∗}(L̃(i_{∗}τ_{F}) ∩ [E]_{Q}) = ch([Γ]_{K}) ∩ [B]_{Q} ∈ H_{4∗}(B; Q) .

Lusztig [19] considered a flat complex vector bundle Γ over an oriented differentiable manifold B with dim(B) = 2j, such that the fibres Γ_{x} (x ∈ B) are nonsingular hermitian forms over C. The twisted signature sign(B, Γ) ∈ Z is defined to be the signature of the nonsingular hermitian form on H^{k}(B; Γ). The complex K-theory signature of Γ is defined by [Γ]_{K} = Γ^{+} − Γ^{−} ∈ KU(B) for any splitting Γ = Γ^{+} ⊕ Γ^{−} with the hermitian form positive/negative definite on Γ^{±}. The twisted signature formula of [19] is

sign(B, Γ) = ⟨ch([Γ]_{K}) ∪ L̃(B), [B]_{Q}⟩ ∈ Z .

Meyer [20] considered a locally constant sheaf Γ over an oriented topological manifold B with dim(B) = 2j, such that the stalks Γ_{x} (x ∈ B) are nonsingular (−)^{k}-symmetric forms over R. The twisted signature sign(B, Γ) ∈ Z is defined to be the signature of the nonsingular symmetric form on H^{k}(B; Γ). The topological K-theory signature of Γ is the topological K-theory signature in the sense of [3] of the flat vector bundle Γ̃ over B with fibres H^{k}(Γ_{x}; R)

[Γ]_{K} = [Γ̃]_{K} ∈ \{ \begin{matrix}KO(B) & if k ≡ 0(mod 2) \\ KU(B) & if k ≡ 1(mod 2) . \end{matrix}

The twisted signature formula of [20] is

sign(B, Γ) = ⟨c̃h([Γ]_{K}) ∪ L(B), [B]_{Q}⟩ ∈ Z

with c̃h = ch ∘ ψ^{2} the modified Chern character obtained by composition with the second Adams operation ψ^{2} and L the original Hirzebruch L-genus defined by

L(B) = ∏_{i=1}^{j} \frac{x_{i}}{tanh x_{i}} ∈ H^{4∗}(B; Q) .

For any complex n-plane bundle α over B with total Chern class

c(α) = ∏_{i=1}^{n}(1 + y_{i}) ∈ H^{2∗}(B)

the Chern characters ch(α) = ∑_{i=1}^{n} e^{y_{i}}, c̃h(α) = ∑_{i=1}^{n} e^{2y_{i}} ∈ H^{2∗}(B; Q) are such that

(c̃h(α) ∪ L(B))_{2j} = (ch(α) ∪ L̃(B))_{2j} ∈ H^{2j}(B; Q) ,

since for any i ≥ 0

\begin{matrix}L(B)_{4i} = 2^{2i−j}L̃(B)_{4i} ∈ H^{4i}(B; Q) , \\ c̃h(α)_{2j−4i} = 2^{j−2i}ch(α)_{2j−4i} ∈ H^{2j−4i}(B; Q) . \end{matrix}

Thus the twisted signature can be expressed as

sign(B, Γ) = ⟨c̃h([Γ]_{K}) ∪ L(B), [B]_{Q}⟩ = ⟨ch([Γ]_{K}) ∪ L̃(B), [B]_{Q}⟩ ∈ Z .

(See ‘Mannigfaltigkeiten und Modulformen’, Bonn notes on lectures of Hirzebruch, pp. 83-4. We are indebted to Michael Crabb for this reference.)

Next, we describe the algebraic L-theory assembly map of Ranicki [32].

Let L. = {L^{d}|d ∈ Z}, L. = {L_{d}|d ∈ Z} be the algebraic L-spectra defined in §13 of [32]. L^{d} is the Kan Δ-set with n-simplexes the d-dimensional symmetric Poincaré n-simplexes over Z, such that

ΩL^{d} = L^{d+1} , π_{∗}(L.) = L^{∗}(Z) ,

and similarly for L_{d}, L. in the quadratic case.

An ‘n-dimensional symmetric cycle’ over a simplicial complex B is an inverse system

(C, ϕ) = {(C[τ], ϕ[τ]) | τ ∈ B}

of (n + |τ| − k)-dimensional symmetric (k − |τ|)-simplexes over Z, with the support {τ ∈ B| C[τ] ≠ 0} contained in a finite subcomplex B_{0} ⊆ B with (k + 2) vertices (so that there exists an embedding B_{0} ⊆ ∂Δ^{k+1}). The cycle is ‘locally Poincaré’ if each (C[τ], ϕ[τ]) (τ ∈ B) is an (n + |τ| − k)-dimensional symmetric Poincaré (k − |τ|)-simplex over Z. The cycle is ‘globally Poincaré’ if the assembly n-dimensional symmetric complex over Z[π_{1}(B)]

(C[B̃], ϕ[B̃]) = ⋃_{τ̃∈B̃}(C[τ], ψ[τ])

is Poincaré, with B̃ the universal cover of B. The generalized homology group H_{n}(B; L.) is identified in §13 of [32] with the cobordism group of locally Poincaré n-dimensional symmetric cycles over B.

The visible symmetric L-group V L^{n}(B) of a simplicial complex B is the cobordism group of globally Poincaré n-dimensional symmetric cycles over B. (For a classifying space B = Bπ these are the original visible symmetric L-groups V L^{∗}(Z[π]) of Weiss [37]). The forgetful maps

\begin{matrix}L_{n}(Z[π_{1}(B)]) ⟶ V L^{n}(B) ; (C, ψ) ⟶ (C, (1 + T)ψ) , \\ V L^{n}(B) ⟶ L^{n}(Z[π_{1}(B)]) ; (C, ϕ) ⟶ (C[B̃], ϕ[B̃]) \end{matrix}

are isomorphisms modulo 8-torsion. Passing from local to global Poincaré cycles defines assembly maps

A : H_{n}(B; L.) ⟶ V L^{n}(B)

for any simplicial complex B. The visible symmetric signature of an n-dimensional geometric Poincaré complex B is an element

σ^{∗}(B) = (C, ϕ) ∈ V L^{n}(B)

with C[τ] = Z (τ ∈ B). The visible symmetric signature of an n-dimensional P L manifold B is the assembly

σ^{∗}(B) = A([B]_{L}) ∈ V L^{n}(B)

of the canonical L.-orientation

[B]_{L} = {(C[τ], ϕ[τ])|τ ∈ B} ∈ H_{n}(B; L.)

with C[τ] = Z.

The generalized homology group H_{n}(B; L.) is the cobordism group of locally Poincaré n-dimensional quadratic cycles over B. The surgery obstruction group L_{n}(Z[π_{1}(B)]) is the cobordism group of globally Poincaré n-dimensional quadratic cycles over B, as well as the cobordism group of n-dimensional quadratic Poincaré complexes over Z[π_{1}(B)]. The quadratic L-theory assembly map

A : H_{n}(B; L.) ⟶ L_{n}(Z[π_{1}(B)]) ; (C, ψ) ⟶ (C[B̃], ψ[B̃])

is defined in §9 of [32] by passing from local to global Poincaré duality. The surgery obstruction of a normal map (f, b) : M → B of closed n-dimensional manifolds is the assembly

σ_{∗}(f, b) = A([f, b]_{L}) ∈ L_{n}(Z[π_{1}(B)])

of an L.-homology surgery invariant [f, b]_{L} ∈ H_{n}(B; L.) with symmetrization

(1 + T)[f, b]_{L} = f_{∗}[M]_{L} − [B]_{L} ∈ H_{n}(B; L.) .

A ‘d-dimensional symmetric Poincaré cocycle’ over a simplicial complex B is a directed system {(C[τ], ϕ[τ]) | τ ∈ B} of (d + |τ|)-dimensional symmetric |τ|-simplexes over Z, i.e. a Δ-map B → L^{d}. The generalized cohomology group H^{−d}(B; L.) = [B, L^{d}] is the cobordism group of d-dimensional symmetric Poincaré cocycles over B.

A d-dimensional symmetric Poincaré cocycle (C, ϕ) over a finite simplicial complex B is homogenous if the structure maps C[σ] → C[τ] (σ ≤ τ ∈ B) are chain equivalances, in which case for any 0-simplex ∗ ∈ B the fundamental group π_{1}(B) acts on the chain homotopy type of the ‘fibre’ d-dimensional symmetric Poincaré complex (C[∗], ϕ[∗]). Let L^{d}(B, Z) be the cobordism group of homogenous d-dimensional symmetric Poincaré cocycles over B, and define an assembly map

A : L^{d}(B, Z) ⟶ L^{d}(π_{1}(B), Z) ; (C, ϕ) ⟶ (C[∗], ϕ[∗]) .

Tensor product over Z makes L. into a ring spectrum, and L. is an L.-module spectrum. The evaluation of the cap product pairing

∩ : H^{r}(B; L.) ⊗ H_{n}(B; L.) ⟶ H_{n−r}(B; L.)

on the canonical L.-coefficient orientation [B]_{L} ∈ H_{n}(B; L.) of an n-dimensional P L manifold B defines L.-coefficient Poincaré duality isomorphisms

− ∩ [B]_{L} : H^{∗}(B; L.) ⟶ H_{n−∗}(B; L.) .

For any simplicial complex B use the cap product pairing

∩ : H^{−d}(B; L.) ⊗ V L^{n}(B) ⟶ V L^{n+d}(B)

and the forgetful map

L^{d}(B, Z) ⟶ H^{−d}(B; L.) ; (C, ϕ) ⟶ (C, ϕ)

to define a product pairing

⊗ : L^{d}(B, Z) ⊗ V L^{n}(B) ⟶ V L^{n+d}(B)

and define similarly the product pairings

\begin{matrix}⊗ : L^{d}(B, Z) ⊗ H_{n}(B; L.) ⟶ H_{n+d}(B; L.) , \\ ⊗ : L^{d}(B, Z) ⊗ H_{n}(B; L.) ⟶ H_{n+d}(B; L.) . \end{matrix}

The products and assembly maps are related by commutative squares

L^{d}(B, Z) ⊗ V L^{n}(B)  ⟶  V L^{n+d}(B)
↓  ↓
L^{d}(π_{1}(B), Z) ⊗ V L^{n}(Z[π_{1}(B)]) ⟶ V L^{n+d}(Z[π_{1}(B)]) ,

L^{d}(B, Z) ⊗ H_{n}(B; L.) ⟶ H_{n+d}(B; L.)
↓  ↓
L^{d}(B, Z) ⊗ V L^{n}(B) ⟶ V L^{n+d}(B) ,
H^{−d}(B; L.) ⊗ H_{n}(B; L.) ⟶  H_{n+d}(B; L.)
↓  ↓
L^{d}(π_{1}(B), Z) ⊗ L_{n}(Z[π_{1}(B)]) ⟶ L_{n+d}(Z[π_{1}(B)]) ,
L^{d}(B, Z) ⊗ H_{n}(B; L.) ⟶  H_{n+d}(B; L.)
↓  ↓
H^{−d}(B; L.) ⊗ L_{n}(Z[π_{1}(B)]) ⟶ L_{n+d}(Z[π_{1}(B)]) .

A fibration F ⟶ E ⟶^{p} B with the fibre F a d-dimensional geometric Poincaré complex induces transfer maps in the quadratic and visible symmetric L-groups

\begin{matrix}p^{∗} : L_{n}(Z[π_{1}(B)]) ⟶ L_{n+d}(Z[π_{1}(E)]) ; (C, ψ) ⟶ (C^{!}, ψ^{!}) \\ p^{∗} : V L^{n}(B) ⟶ V L^{n+d}(E) ; (C, ϕ) ⟶ (C^{!}, ϕ^{!}) \end{matrix}

with C^{!}[τ] = C[pτ] (τ ∈ B). In general, there is no transfer p^{∗} : L^{n}(Z[π_{1}(B)]) → L^{n+d}(Z[π_{1}(E)]) in the symmetric L-groups (cf. Appendix 2 of [18]).

A P L fibre bundle F ⟶ E ⟶^{p} B with the fibre a d-dimensional P L manifold F induces transfer maps in L.-homology

\begin{matrix}p^{∗} : H_{n}(B; L.) ⟶ H_{n+d}(E; L.) , \\ p^{∗} : H_{n}(B; L.) ⟶ H_{n+d}(E; L.) \end{matrix}

which commute with the assembly maps

\begin{matrix}H_{n}(B; L.) & \underrightarrow{A} & V L^{n}(B) \\ _{p^{∗} ↓} & & _{↓ p^{∗}} \\ H_{n+d}(E; L.) & \underrightarrow{A} & V L^{n+d}(E) , \end{matrix} ; \begin{matrix}H_{n}(B; L.) & \underrightarrow{A} & L_{n}(Z[π_{1}(B)]) \\ _{p^{∗} ↓} & & _{↓ p^{∗}} \\ H_{n+d}(E; L.) & \underrightarrow{A} & L_{n+d}(Z[π_{1}(E)]) . \end{matrix}

The π_{1}(B)-equivariant symmetric signature is the assembly

σ^{∗}(F, p) = A([F, p]_{L}) ∈ L^{d}(π_{1}(B), Z)

of the L.-coefficient fibre transport

(F, p)_{L} ∈ L^{d}(B, Z)

represented by the Δ-map

(F, p)_{L} : B ⟶ L^{d} ; τ ⟶ (C(p^{−1}(τ)), ϕ_{τ}) .

The proof is by a direct generalization of the expression in §16 of [32] of the visible symmetric signature of F as the assembly of the canonical L-orientation. Let [F, p]_{L} ∈ H^{−d}(B; L.) be the image of (F, p)_{L} ∈ L^{d}(B, Z), such that

p_{∗} ∘ p^{∗} = [F, p]_{L} ⊗ − : H_{n}(B; L.) ⟶ H_{n+d}(B; L.) ,
p_{∗} ∘ p^{∗} = [F, p]_{L} ⊗ − : V L^{n}(B) ⟶ V L^{n+d}(B) ,
p_{∗} ∘ p^{∗} = [F, p]_{L} ⊗ − : L_{n}(Z[π_{1}(B)]) ⟶ L_{n+d}(Z[π_{1}(B)]) .

This gives an alternative proof of 2.7, at least in outline.

For a P L fibre bundle F ⟶ E ⟶^{p} B with base an n-dimensional P L manifold B and fibre a d-dimensional P L manifold F the total space is an (n + d)-dimensional P L manifold E. The tangent bundle of E is a direct sum τ_{E} = i_{∗}τ_{F} ⊕ p^{∗}τ_{B} (as in the differentiable case). Let (E(i_{∗}τ_{F}), S(i_{∗}τ_{F})) be the (n + 2d)-dimensional manifold with boundary defined by the total space of the (D^{d}, S^{d−1})-bundle of the d-plane bundle i_{∗}τ_{F} over E. The L.-coefficient Thom class of i_{∗}τ_{F} is an element U_{i_{∗}τ_{F}} ∈ Ḣ^{d}(T (i_{∗}τ_{f}); L.), with Ḣ^{∗} reduced cohomology and T (i_{∗}τ_{F}) = E(i_{∗}τ_{F})/S(i_{∗}τ_{F}) the Thom space. The Poincaré duality isomorphism

\begin{matrix}[E(i_{∗}τ_{F})]_{L} ∩ − : Ḣ^{d}(T (i_{∗}τ_{f}); L.) = H^{d}(E(i_{∗}τ_{f}), S(i_{∗}τ_{F}); L.) \\ ⟶ H_{n+d}(E(i_{∗}τ_{f}); L.) = H_{n+d}(E; L.) \end{matrix}

is such that

[E(i_{∗}τ_{F})]_{L} ∩ U_{i_{∗}τ_{F}} = [E]_{L} ∈ H_{n+d}(E; L.) .

The canonical L.-orientation [E]_{L} of E is the transfer of [B]_{L} ∈ H_{n}(B; L.)

[E]_{L} = p^{∗}[B]_{L} ∈ H_{n+d}(E; L.) ,

and

p_{∗}[E]_{L} = p_{∗}p^{∗}[B]_{L} = [F, p]_{L} ∩ [B]_{L} ∈ H_{n+d}(B; L.)

is the Poincaré dual of the fibre transport [F, p]_{L} ∈ H^{−d}(B; L.). The transfer map in L.- cohomology

p^{!} : H^{0}(E; L.) ≅ H_{n+d}(E; L.) ⟶^{p_{∗}} H_{n+d}(B; L.) ≅ H^{−d}(B; L.)

is such that

p^{!}(1) = [F, p]_{L} ∈ H^{−d}(B; L.) .

This is an L-theoretic analogue of the result of Becker and Schultz ([4], Section 6) that for a differentiable fibre bundle the transfer map p^{!} : KO(E) → KO(B) in real K-theory is such that

p^{!}(1) = ∑_{i}(−1)^{i}B̃ ×_{π_{1}(B)} H^{i}(F; R) ∈ KO(B)

with B̃ the universal cover of B.

Rationally, L. has the homotopy type of a product of Eilenberg-MacLane spectra

L. ⊗ Q = ∏_{i=0}^{∞} K(Q, 4i) .

The canonical L.-orientation [B]_{L} ∈ H_{n}(B; L.) of an oriented n-dimensional manifold B is such that

[B]_{L} ⊗ 1 = L(B) ∩ [B]_{Q} ∈ H_{n}(B; L.) ⊗ Q = H_{n−4∗}(B; Q)

with [B]_{Q} ∈ H_{n}(B; Q) the ordinary fundamental class.

A homogenous 2k-dimensional symmetric Poincaré cocycle (C, ϕ) over a finite simplicial complex B determines a representation of π_{1}(B) on H^{k}(C[∗]; R) preserving the nonsingular (−)^{k}-symmetric form over R. Working as in Meyer [20] define a flat vector bundle over B with fibre H^{k}(C[∗]; R)

Γ = B̃ ×_{π_{1}(B)} H^{k}(C[∗]; R)

equipped with an automorphism A : Γ → Γ such that A^{2} = (−)^{k}I. For k ≡ 0(mod 2) the ±1-eigenspaces of A are the bundles Γ^{±} = B̃ ×_{π_{1}(B)} H^{k}(C[∗]; R)^{±} over B with fibres complementary subspaces H^{k}(C[∗]; R)^{±} ⊆ H^{k}(C[∗]; R) where the symmetric form is positive/negative definite, and the real K-theory signature of (C, ϕ) is defined by

[C, ϕ]_{K} = [Γ^{+}] − [Γ^{−}] ∈ KO(B) .

For k ≡ 1(mod 2) A defines a complex structure on Γ, and the complex K-theory signature of (C, ϕ) is defined by

[C, ϕ]_{K} = [Γ^{∗}] − [Γ] ∈ KU(B)

with Γ^{∗} the complex conjugate bundle. The topological K-theory signature defines morphisms

L^{2k}(B, Z) → K(B) ; (C, ϕ) ⟶ [C, ϕ]_{K}

(where K = KO or KU) such that there is defined a commutative diagram

L^{2k}(B, Z) ⟶  H^{−2k}(B; L.)
↓  ↓
K(B)  \underrightarrow{c̃h} H^{−2k+2∗}(B; Q) .

Given a P L fibre bundle F ⟶ E ⟶^{p} B with the fibre a F a 2k-dimensional manifold let (C, ϕ) be the homogenous 2k-dimensional symmetric Poincaré cocycle representing the L.-coefficient fibre transport (F, p)_{L} ∈ L^{2k}(B, Z) with

C[τ] = C(p^{−1}(τ)) (τ ∈ B) , H^{∗}(C[∗]; R) = H^{∗}(F; R) .

The topological K-theory signature of the bundle Γ = B̃ ×_{π_{1}(B)} H^{k}(F; R) over B is denoted by

[F, p]_{K} = [Γ]_{K} ∈ K(B) .

By the above, the rationalization of [F, p]_{L} ∈ H^{−2k}(B; L.) is identified with the Chern character of [F, p]_{K}

[F, p]_{L} ⊗ 1 = c̃h([F, p]_{K}) ∈ H^{−2k}(B; L.) ⊗ Q = H^{−2k+4∗}(B; Q) ,

and so p_{∗} ∘ p^{∗} = c̃h([F, p]_{K}) ⊗ − : H_{∗}(B; L.) ⊗ Q ⟶ H_{∗+2k}(B; L.) ⊗ Q ,

p_{∗} ∘ p^{∗} = c̃h([F, p]_{K}) ⊗ − : V L^{∗}(B) ⊗ Q = L_{∗}(Z[π_{1}(B)]) ⊗ Q

⟶ V L^{∗+2k}(B) ⊗ Q = L_{∗+2k}(Z[π_{1}(B)]) ⊗ Q .

Let now F ⟶ E ⟶^{p} B be a P L fibre bundle with the fibre F a 2k-dimensional manifold and the base B an n-dimensional manifold, so that the total space E is an (n + 2k)-manifold. The canonical L.-orientation [E]_{L} ∈ H_{n+2k}(E; L.) of E has image p_{∗}[E]_{L} ∈ H_{n+2k}(B; L.) with rationalization

p_{∗}[E]_{L} ⊗ 1 = ([F, p]_{L} ∩ [B]_{L}) ⊗ 1
= p_{∗}(L(E) ∩ [E]_{Q})
= c̃h([F, p]_{K}) ∩ (L(B) ∩ [B]_{Q}) ∈ H_{n+2k−∗}(B; Q) .

For n + 2k ≡ 0(mod 4) the signature of E is given by

sign(E) = ⟨L(E), [E]_{Q}⟩
= ⟨L(E), p^{∗}[B]_{Q}⟩
= ⟨p_{∗}L(E), [B]_{Q}⟩
= ⟨c̃h([F, p]_{K}) ∪ L(B), [B]_{Q}⟩
= ⟨ch([F, p]_{K}) ∪ L̃(B), [B]_{Q}⟩ ∈ Z

as in [3]. The degree 0 component of ch([F, p]_{K}) ∈ H^{−2k+4∗}(B; Q) is the rank of the virtual bundle [F, p]_{K}, which is sign(F) ∈ Z ⊂ H^{0}(B; Q) = Q (to be interpreted as 0 if k ≡ 1(mod 2)). If all the other components are 0 then the signature is multiplicative

sign(E) = sign(F) sign(B) ∈ Z ,

and more generally p_{∗} ∘ p^{∗} = sign(F) ⊗ − : H_{∗}(B; L.) ⊗ Q ⟶ H_{∗+2k}(B; L.) ⊗ Q ,

p_{∗} ∘ p^{∗} = sign(F) ⊗ − : V L^{∗}(B) ⊗ Q = L_{∗}(Z[π_{1}(B)]) ⊗ Q
⟶ V L^{∗+2k}(B) ⊗ Q = L_{∗+2k}(Z[π_{1}(B)]) ⊗ Q .

See Neumann [26] for a class of groups G such that ch([F, p]_{K}) is concentrated in degree 0 if the action of π_{1}(B) on H^{k}(F; C) factors as

ω : π_{1}(B) ⟶ G ⟶ Aut H^{k}(F; C) .

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Current addresses

Wolfgang Lück Andrew Ranicki Department of Mathematics Department of Mathematics University of Kentucky University of Edinburgh 735 Patterson Office Tower James Clerk Maxwell Building Lexington, Kentucky 40506 Edinburgh EH9 3JZ U.S.A Scotland