UNIVERSAL GEOMETRIC EXAMPLES FOR TRANSFER MAPS IN ALGEBRAIC K AND L-THEORY

ERIK KJÆR PEDERSEN

0. Introduction

In this paper we will be concerned with various geometrically defined transfer maps in algebraic K-theory. In [3], Ehrlich shows that given a fibration F ⟶^{i} E ⟶^{p} B with finitely dominated fiber and base, there is a homomorphism p^{∗} : K̃_{0}(Zπ_{1}B) → K̃_{0}(Zπ_{1}E) which is related to the finiteness obstruction of E, σ(E) by the formula: σ(E) = p^{∗}(σ(B)) + i_{∗}(σ(F) · χ(B), where χ(B) denotes the Euler characteristic of B.

Similarly, given a PL bundle Anderson [2] defines a homomorphism p^{∗} : W̃h(Zπ_{1}B) → W̃h(Zπ_{1}E) which, given a fiber homotopy equivalence

\xymatrix{F′ \ar[d] \ar[r]^{f} & F \ar[d] \\ E′ \ar[d] \ar[r]^{g} & E \ar[d] \\ B′ \ar[r]^{h} & B}

relates the torsion of the maps by the analogous formula: τ(g) = p^{∗}(τ(h)) + i_{∗}(τ(f)) · χ(B).

A final example to have in mind is the transfer like homomorphism in Wall’s L-groups coming from bundles: Given a topological bundle F → E → B with F a topological manifold, there is a transfer homomorphism L_{i}(Zπ_{1}B) → L_{i+dim M}(Zπ_{1}E) which relates surgery obstructions of base and total space in a bundle, see [11]. In case of S^{1}-bundles Wall shows [11, p. 123] that this transfer only depends on the fundamental groups in the bundle and the orientation, in particular it does not depend on π_{2} of the base. His method is to construct a universal S^{1}-bundle with given fundamental group data.

In this paper we generalize this result to arbitrary fibrations (bundles) F → E → B. We construct universal fibrations with given fundamental group data (see Section 1 for precise definition) such that a given fibration is the pullback of this universal fibration via a map inducing isomorphisms of fundamental groups. In course of the construction we determine which fundamental group sequences are possible for a fibration with fiber F, in terms of G_{1}(F), a subgroup of π_{1}(F) (see [4]). We apply the construction to show the following theorem.

Partially supported by the Danish Research Council.

Theorem. The geometrically defined K̃_{0}, Wh and L-group transfer maps of a fibration, respectively PL-bundle, respectively topological bundle F → E → B, only depend on F and the exact sequence π_{1}F → π_{1}E → π_{1}B, and the orientation, that is π_{1}(E) → π_{0}(H(F, ∗)) (H(F, ∗)= basepoint preserving homotopy equivalences) in the K̃_{0}-case, π_{1}(E) → π_{0}(PL(F, ∗)) in the Wh-case and π_{1}(E) → π_{0}(Homeo(F, ∗)) in the L-group case.

Remark. The K̃_{0}-case was proved by Ehrlich in [3] for fibrations with π_{1}(B) → π_{0}(H(F)) trivial, and conjectures in general (H(F) = unbased homotopy equivalences of F).

The results facilitate computation of the transfer maps in a number of interesting cases. We exemplify this by giving an algebraic description of the K̃_{0} and Wh-transfer of S^{1}-fibrations, whenever the extension Z/n → π_{1}E → π_{1}B determined by the fibration is the reduction mod n of some integral extension Z → ρ → π_{1}B. In this algebraic description we use results of [8] and [9].

1. General notation

Let F be a fixed connected space of the homotopy type of a countable CW-complex. We will be concerned with fibrations with fibre F. Such fibrations have a universal classifying fibration F → EF → BF see [1]. We shall also be concerned with fibrations with additional structure, e. g. topological bundles or PL-bundles, but always of a kind that have classifying spaces. Using the above notation we may readily think of EF and BF as classifying spaces of some restricted kind of fibrations as well. However in the case of fibrations if we denote the monoid of homotopy equivalences of F by H(F) and basepoint preserving homotopy equivalences H(F, ∗), we may reinterpret EF and BF as BH(F, ∗) and BH(F) respectively. The boundary map in homotopy π_{i}BF → π_{i−1}F then corresponds to the map induced by evaluation at basepoint π_{i−1}(H(F)) → π_{i−1}(F). In [4], Gottlieb defines and studies the subgroup G_{1}(F) ⊂ π_{1}(F) given by G_{1}(F) = Im(π_{2}BF → π_{1}F). Among other things it is shown in [4] that G_{1}(F) is contained in Cπ_{1}F, the center of π_{1}F. By the fundamental group data of a fibration F → Y → X we will understand the exact sequence 0 → A → π_{1}F → π_{1}Y → π_{1}X together with the homomorphism induced by classification π_{1}Y → π_{1}EF which since π_{1}EF = π_{0}(H(F, ∗)) records the way π_{1}Y acts on the fiber and hence also the action of π_{1}X on the fiber. In the exact sequence above A denotes Im(π_{2}X → π_{1}F).

Two fibrations F → Y_{i} → X_{i}, i = 1, 2, are said to have isomorphic fundamental group data if there are isomorphisms π_{1}(Y_{1}) ≅ π_{1}(Y_{2}) and π_{1}(X_{1}) ≅ π_{1}(X_{2}) making the diagrams

[figure: 0; A_{1}; π_{1}(F); π_{1}(Y_{1}); ≅; π_{1}(X); ≅; 1; 0; A_{2}; π_{1}(F); π_{1}(Y_{2}); π_{1}(X_{2}); 1]

and

\xymatrix{π_{1}(Y_{1}) \ar[dr]_{≅} \\  & π_{1}(EF) \\ π_{1}(Y_{2}) \ar@{-->}[ur]}

commutative. The main aim of this paper is to construct universal fibrations with given fundamental group data. However, the methods are general obstruction theory, so if we think of F → EF → BF as e. g. the universal topological bundle with fibre F we also obtain universal bundles with given fundamental group data, so even though, throughout the paper, we will use the language of fibrations, in the applications we shall need the results for fibrations, as well as PL and topological bundles.

Throughout the paper we will use K(π, n) to denote an Eilenberg–MacLane space of type (π, n). The letter B in front of a space will denote the classifying space of fibrations, thus BK(π, 1) will be the classifying space of fibrations with fiber K(π, 1). If π is a group we use Aut(π) to denote the group of automorphisms of π and Out(π) the group of outer automorphisms.

2. The obstruction to group extensions reinterpreted

Given discrete groups G and π and a homomorphism π → Out(G) there is an obstruction in H^{3}(K(π, 1); CG) (local coefficients) to the existence of an exact sequence 1 → G → ρ → π → 1 inducing the given action of π on G. If this obstruction vanishes such extensions are classified up to isomorphism by H^{2}(K(π, 1); CG)[5]. On the other hand the above exact sequence does imply there is a fibration K(G, 1) → K(ρ, 1) → K(π, 1). We want to relate the extension obstruction to the problem of constructing such a fibration. This involves a study of fibrations with fibre K(G, 1). Such a study is done in [7] in the case G is abelian. However, using the same kind of semisimplicial methods it is clear that we may take BK(G, 1) to be the realization of the classifying space of the semisimplicial group of automorphisms of K(G, 1) where K(G, 1) is the standard semisimplicial model (essentially the bar construction). Similarly EK(G, 1) is obtained from basepoint preserving automorphisms of K(G, 1).

Proposition 2.1. In the fibration K(G, 1) → EK(G, 1) → BK(G, 1) we have EK(G, 1) = B Aut G, BK(G, 1) has only two nonzero homotopy groups, Out G and CG.

Proof. We use the model for K(G, 1) with K(G, 1)_{n} = G × G × ⋯ G, n factors, and

∂_{0}(x_{1}, …, x_{n}) = (x_{2}, …, x_{n}),
∂_{i}(x_{1}, …, x_{n}) = (x_{1}, …, x_{i}x_{i+1}, …, x_{n}),
∂_{n}(x_{1}, …, x_{n}) = (x_{1}, …, x_{n−1},

and degeneracies inserting 1’s at appropriate places. Using [7] one sees that an n-simplex in the mapping space Hom(K(G, 1), K(G, 1)) is a collection of maps f_{ij} : G → G, 0 ≤ i ≤ j ≤ n, where f_{ii} is a group homomorphism and f_{ij} is left f_{ii} and right f_{jj} equivariant, i. e. f_{ij}(a · b) = f_{ii}(a) · f_{ij}(b) = f_{ij}(a) · f_{jj}(b). We get the automorphisms by requiring all f_{ij} to be 1 − 1 and onto. We have the relation of f_{jj}(a) = f_{ij}(1)^{−1} · f_{ii}(a) · f_{ij}(1) from which it is seen that π_{0} is automorphisms modulo inner automorphisms. It also shows that π_{1} is the center of G since f_{00} = f_{11} = 1_{G} implies that f_{01}(1) ∈ CG and this determines π_{1}. The fixed point preserving automorphisms are the ones satisfying f_{ij}(1) = 1 for all i ≤ j which shows that f_{ii} = f_{ij} = f_{jj}, so the fixed point preserving automorphisms are exactly Aut G. The results now follow. □

Corollary 2.2. There is up to homotopy a fibration K(CG, 2) → BK(G, 1) → K(Out G, 1). Given a homomorphism π → Out G we may identify the above mentioned obstruction to construct an exact sequence, with the obstruction to solve the lifting problem

\xymatrix{ & BK(G, 1) \ar[d] \\ K(π, 1) \ar@{-->}[ur] \ar[r] & K(Out G, 1)}

and if this obstruction vanishes, the set of liftings classified by H^{2}(K(π, 1); CG) corresponds to the classification of extensions.

Proof. Using the models for HK(π, 1) of the above proof, the considerations in [5, p. 124] translate directly to the lifting problem. □

3. Fibrations with given fundamental group data

Given a fibration F → Y → X one immediately sees that Im(π_{2}X → π_{1}F) = A is a π_{1}X invariant subgroup contained in G_{1}F. If on the other hand we have a group π, a homomorphism π → Out(π_{1}F), and a subgroup A of π_{1}F invariant under this action, we may ask if there is a fibration realizing this data. The first condition that has to be fulfilled is that the map π → Out(π_{1}F) factors π → π_{0}(H(F)) → Out π_{1}F. If this is the case we choose such a factorization and fix it. We will now replace BF which has π_{1}BF = π_{0}(H(F)) by some cover of BF with fundamental group Im(π → π_{0}(H(F))). We pull back the universal fibration to this cover and get a universal fibration with some restriction on the way π_{1} of the base is allowed to act on the fibre. We will still denote this by F → EF → BF.

To study the question of realizing fibrations with given properties of the fundamental groups we need to construct a space BF (A) by the following procedure: Consider the canonical homomorphism π_{2}BF → G_{1}F/A. The kernel is π_{1}BF invariant so we may attach 3-cells to kill this kernel. We proceed to attach higher dimensional cells to kill higher homotopy groups.

Lemma 3.1. Up to canonical homotopy equivalence BF (A) is universal with respect to the properties

(a) π_{i}(BF (A)) = 0 for i ≥ 3.

(b) π_{2}(BF (A)) = G_{1}F/A.

(c) π_{1}BF (A)) = π_{1}BF.

(d) There is a map BF → BF (A) inducing the canonical maps on π_{1} and π_{2}.

Proof. Clearly BF (A) satisfies conditions (a) through (d). Let X be some space satisfying the conditions; then we have

\xymatrix{BF \ar[rr] \ar[dr] &  & BF (A) \ar@{-->}[dl] \\  & X}

and using elementary obstruction theory there is a unique homotopy class of maps we can fill in to make the diagram homotopy commutative. This map must induce isomorphism in homotopy so it is a homotopy equivalence. We now construct a diagram

\xymatrix{F \ar[d] \ar[r] & K(π_{1}F/A, 1) \ar[r] \ar[d] & K(π_{1}F/A.1) \ar[d] \\ EF \ar[d] \ar[r] & K(π_{1}EF, 1) \ar[d] \ar[r] & EK(π_{1}F/A, 1) \ar[d] \\ BF \ar[r] & BF (A) \ar[r] & BK(π_{1}F/A, 1)}  (*)

Here all vertical sequences are fibrations up to homotopy. The map EF → K(π_{1}EF, 1) classifies the universal cover of EF. The map K(π_{1}EF, 1) → BF (A) is seen to exist and to be unique up to homotopy to make the diagram commutative. That the homotopy fibre of K(π_{1}EF, 1) → BF (A) is a K(π_{1}F/A, 1) follows from the exact sequence

[figure: 0; G_{1}F/A; π_{1}F/A; π_{1}EF; π_{1}BF; 1; π_{2}BF (A); π_{1}BF (A)]

Finally BF (A) → BK(π_{1}F/A, 1) is the classifying map. We notice that the homotopy exact sequences may be thought of as follows:

\xymatrix{0 \ar[r] & G_{1}F/A \ar@{^{(}->}[d] \ar[r] & π_{1}F/A \ar[d] \ar[r] & π_{0}(H(F, ∗)) \ar[r] \ar[d] & π_{0}H(F) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & Cπ_{1}F/A \ar[r] & π_{1}F/A \ar[r] & Aut π_{1}F/A \ar[r] & Out(π_{1}F/A) \ar[r] & 0}

Here we identify π_{0}(H(F)) with π_{1}BF and π_{0}(H(F, ∗)) with π_{1}EF and remember the remarks in the beginning of this section that imply we do not have an induced homomorphism π_{0}(H(F, ∗)) → Aut(π_{1}F/A). □

From the diagram (∗) we may construct a new diagram

\xymatrix{K(G_{1}F/A, 2) \ar[d] \ar[r] & K(Cπ/A, 2) \ar[d] \\ BF (A) \ar[d] \ar[r] & BK(π_{1}F/A, 1) \ar[d] \\ K(π_{1}(BF), 1) \ar[r] & K(Out(π_{1}F/A), 1)}  (**)

Proposition 3.2. Given a group π, a homomorphism π → π_{1}BF and a subgroup A ⊂ G_{1}(F) invariant under the induced action of π, there is an obstruction p ∈ H^{3}(K(π, 1); G_{1}F/A) (local coefficients) to the existence of a fibration F → Y → X with π_{1}X ≅ π, Im(π_{2}X → π_{1}F) = A and the action of π_{1}X on F given by π_{1}X ≅ π → π_{1}BF. The image of this obstruction in H^{3}(K(π, 1); Cπ_{1}F/A) is the obstruction of [5] to realize an exact sequence π_{1}F/A → ρ → π with π acting on π_{1}F/A through the given π → π_{1}BF → Out(π_{1}F/A).

Proof. The homomorphism π → π_{1}BF induces a map K(π, 1) → K(π_{1}BF, 1). We define p ∈ H^{3}(K(π, 1); G_{1}F/A) to be the obstruction to lift this map to BF (A) in the diagram (∗∗). The final remarks on the image of p in H^{3}(K(π, 1); π_{1}F/A) now follow from the diagram (∗∗), naturality of obstructions and Section 2. If this obstruction vanishes we choose a lift l and construct a pullback diagram defining BF (A, l):

\xymatrix{BF (A, l) \ar[d] \ar[r] & K(π, 1) \ar[d] \\ BF \ar[r] & BF (A)}

The pullback of the universal fibration F → EF → BF to BF (A, l) we denote F → EF (A, l) → BF (A, l) and we claim this fibration satisfies the assumptions. To see this consider the homotopy fibre H of K(π, 1) → BF (A, l). We have the exact sequence 0 → G_{1}F/A → π_{1}H → π → π_{1}BF → 1 to compute the homotopy groups of H, so H is a K(π_{1}H, 1). This implies the homotopy fibre of BF (A, l) → BF is a K(π_{1}H, 1), and if we compare exact sequences

[figure: 0; π_{2}(BF (A, l)); π_{2}BF; π_{1}H; π_{1}BF (A, l); π_{1}BF; 1; 0; 0; G_{1}F/A; π_{1}H; π; π_{1}BF; 1]

we see, using some version of the five lemma, that π_{1}(BF (A, l)) ≅ π by an isomorphism making π_{1}(BF (A, l)) → π_{1}BF the given map π → π_{1}BF, and further that the sequence 0 → π_{2}(BF (A, l)) → π_{2}BF → G_{1}F/A → 0 is exact. This implies that Im(π_{2}(BF (A, l)) → π_{1}F) = A and we have constructed a fibration with the sought properties.

On the other hand if such a fibration F → Y → X does exist, we must prove that the obstruction vanishes. We have the following diagram

\xymatrix{F \ar[r] \ar[d] & F \ar[d] \\ Y \ar[d] \ar[r] & EF \ar[d] \\ X \ar[dr] \ar[r] & BF \ar[r] & BF (A) \\  & K(π, 1) \ar@{-->}[ur]}

We may think of K(π, 1) as X with 3-cells attached to kill π_{2}(X) etc. and the map X → K(π, 1) as inclusion. By assumption, the image of π_{2}(X) in π_{2}(BF) is the kernel π_{2}BF → G_{1}F/A so the map π_{2}X → π_{2}(BF (A)) = G_{1}F/A is trivial so there is no obstruction to extend the map to the 3-skeleton. Since higher homotopy groups of BF (A) vanish we meet no further obstructions. We also notice that the homotopy class of the extension K(π, 1) → BF (A) is uniques (using obstruction theory once again) and that it is indeed a lift of the given map K(π, 1) → K(π_{1}BF, 1). □

Remark 3.3. If we have a fibration F → Y → X with fundamental group data 0 → A → π_{1}(F) → π_{1}(Y) → π_{1}X → 1 and π_{1}Y → π_{1}EF, it follows easily from the proof of proposition 3.2 that the exact sequences

π_{1}F/A → ρ → π_{1}X

we may obtain from fibrations, with π_{1}X → Out(π_{1}F/A) induced by π_{1}Y → π_{1}EF, are given by the action of H^{2}(K(π_{1}X, 1); G_{1}F/A) on the exact sequence π_{1}F/A → π_{1}(Y) → π_{1}X through the standard action of H^{2}(K(π_{1}X, 1); (π_{1}F/A)), see [5].

4. The universal fibrations

In the proof of Proposition 3.2 we constructed spaces BF (A, l) as the pullback of the diagram

\xymatrix{BF (A, l) \ar[r] \ar[d] & K(π, 1) \ar[d]^{l} \\ BF \ar[r] & BF (A)}

where l is a lift of a given map K(π, 1) → K(π_{1}BF, 1). We also saw that a fibration F → Y → X with A = Im(π_{2}X → π_{1}F) determines a unique lift l : K(π_{1}X, 1) → BF (A), and we thus get a pullback diagram

[figure: F; F; Y; EF (A, l); X; BF (A, l)]

It is easily seen that the horizontal maps induce isomorphisms of fundamental group data. To see that F → EF (A, l) → BF (A, l) may be considered a universal fibration with the given fundamental group data, it thus suffices to show that two fibrations with isomorphic fundamental group data give rise to the same lift l : K(π, 1) → BF (A). Let us assume we have two fibrations with isomorphic fundamental; group data F → Y_{i} → X_{i}, i = 1, 2. Denote the common fundamental group of X_{i} by π. We obtain diagrams of fibrations

\xymatrix{F \ar[d] \ar[r] & K(π_{1}F/A, 1) \ar[d] \\ Y_{i} \ar[r] \ar[d] & K(π_{1}Y_{i}, 1) \ar[d] \\ X_{i} \ar[r] & K(π, 1)}

and a pullback diagram

\xymatrix{K(π_{1}F/A, 1) \ar[d] & = & K(π_{1}F/A, 1) \ar[d] & = & K(π_{1}F/A, 1) \ar[d] \\ K(π_{1}Y_{i}, 1) \ar[d] \ar[rr] &  & K(π_{1}EF, 1) \ar[d] \ar[rr] &  & EK(π_{1}F/A, 1) \ar[d] \\ K(π, 1) \ar[rr]^{l_{i}} &  & BF (A) \ar[rr] &  & BK(π_{1}F/A, 1)}

The composites K(π, 1) ⟶^{l_{i}} BF (A) → BK(π_{1}F/A, 1), i = 1, 2 are homotopic since we have isomorphic fundamental group sequences. We may thus think of l_{1} and l_{2} as lifts of the same map into BK(π_{1}F/A, 1) covering given maps into K(π_{1}BF, 1). The difference between l_{1} and l_{2} is by obstruction theory given by an element of H^{1}(K(π, 1); Cπ_{1}F/G_{1}F). However, the maps K(π_{1}Y_{i}, 1) → K(π_{1}EF, 1) are by assumption homotopic and since they are determined by the fundamental groups they are homotopic covering the map into EK(π_{1}F/A, 1). We thus obtain that the difference between l_{1} and l_{2} in H^{1}(K(π, 1); Cπ_{1}F/G_{1}F) maps to 0 in H^{1}(K(π_{1}Y_{i}, 1); Cπ_{1}F/G_{1}F). This map however is a monomorphism since π_{1}Y_{i} → π is onto so l_{1} is homotopic to l_{2} and we have proved the following:

Theorem 4.1. Suppose there exists a fibration X → Y with fibre F whose projection induces the homomorphism f : A → B of fundamental groups and ϕ : A → π_{1}EF describing the action of A on the fiber. Then there exists a universal fibration P : Z → W so that given any fibration p : E → M with fiber F and isomorphisms h_{1} : π_{1}E → A, h_{2} : π_{1}(M) → B with fh_{1} = h_{2}π_{1}(p) and ϕ · h_{1} the classifying map π_{1}E → π_{1}EF, there exists a unique homotopy class of fibrewise maps of p to P inducing the stated isomorphisms.

Remark. This theorem was originally proved in [11, p. 123] for the case S^{1} and S^{0} fibrations.

Remark. Completely analogous theorems hold if we replace the word fibration with topological bundle, PL-bundle or any other kind of bundle theory, the proof being verbatim the same. Of course in the case of e. g. topological bundles one replaces π_{0}H(F) by π_{0}(Homeo(F)), G_{1}(F) by Im(π_{1}(Homeo F) → π_{1}F) etc.

5. Application to geometrically defined transfers

In this section we apply the results of Section 4 to various geometrically defined transfer maps discussed in the introduction. The transfer map of Wall groups is defined for any topological bundle with fibre a topological manifold [11], whereas the K_{0} and Wh transfers are only defined for respectively fibrations and PL-bundles with finite base space. The first order of the day is to remedy this situation.

Proposition 5.1. Given a Hurewicz fibration F → E → B with F finitely dominated and π_{1}B finitely presented. There exists a (transfer) homomorphism p^{∗} : K̃_{0}(Zπ_{1}B) → K̃_{0}(Zπ_{1}E) uniquely determined by the following properties:

(1) p^{∗} is natural with respect to pullbacks.

(2) If B is finitely dominated, p^{∗} agrees with the transfer map of [3] mentioned in the introduction.

Proof. If B is finitely dominated we let p^{∗} agree with Ehrlich’s definition. If B is not finitely dominated, we may find a finite complex K and a map K → B inducing isomorphism on the fundamental group and such that π_{2}K → π_{2}B/ ker(π_{2}B → π_{1}F) is onto. This may be done by first constructing K using a given presentation and then take a wedge product with finitely many 2-spheres. We obtain a pullback fibration

[figure: F; F; h^{∗}(E); E; K; B]

and h^{∗}(E) → E as well as K → B induce isomorphisms on π_{1}. We may now define p^{∗} to be the composite

K̃_{0}(Zπ_{1}B) ⟶^{≅} K̃_{0}(Zπ_{1}K) ⟶^{p_{K}^{∗}} K̃_{0}(Zπ_{1}h^{∗}E) ⟶^{≅} K̃_{0}(Zπ_{1}E).

To see this is well defined let (K_{i}, h_{i}), i = 1, 2, be as above. We then form

\xymatrix{K_{1} ∨ K_{2} \ar@{^{(}->}[d] \ar[r] & B \\ X \ar[ur]}

where X is obtained from K_{1} ∨ K_{2} by attaching 2-cells to K_{1} ∨ K_{2} to make generators of π_{1}(K_{1}) equal to the corresponding elements in π_{1}(K_{2}). This makes X satisfy the conditions above and by naturality of Ehrlich’s transfer the transfers defined using (K_{1}, h_{1}) respectively (K_{2}, h_{2}) will coincide. Naturality is proved similarly using naturality of the transfer defined by Ehrlich. □

The corresponding extension of the Wh-transfer of arbitrary PL-bundles is proved entirely similarly once it is noticed that the transfer map defined by Anderson in [2] actually is natural with respect to pullback.

Applying the transfer maps to the universal fibrations (PL-bundles, Top-bundles) we obtain the following.

Theorem 5.2. Given fibrations (PL-bundles, Top-bundles), the transfer maps in K̃_{0} (Wh, L_{∗}) discussed in the introduction only depend on the fundamental group data of the fibration (PL-bundle, Top-bundle).

Remarks 5.3. In the case of S^{1}-bundles the above theorem is proved for the L-groups in [11] using this method. In [3] Ehrlich proved the theorem for fibrations with π_{1} of the basespace acting trivial on the fibre for the K̃_{0}-transfer. This theorem is a new proof of that, but with no assumptions on the action of π_{1} (as was conjectured in [3]).

We obtain a number of corollaries of this theorem. We shall concentrate on applications to the K̃_{0}-transfer and only mention the modifications needed in other cases.

Corollary 5.4. Consider a fibration F → X → Y with finitely dominated fibre F and fundamental group data

\xymatrix{π_{1}F/A \ar[d] \\ ρ \ar[d] \ar[r] & π_{1}EF \\ π}  (*)

and let B ⊆ G_{1}F be some π_{1}BF invariant subgroup of G_{1}F with A ⊆ B. Then there exists a fibration with fundamental group data

\xymatrix{π_{1}F/B \ar[d] \\ ρ/(B/A) \ar[d] \ar[r] & π_{1}EF \\ π}  (**)

and the associated transfer map is K̃_{0}(Zπ) → K̃_{0}(Zρ) → K̃_{0}(Z(ρ/(B/A))).

Proof. We may find a map from the wedge of S^{2}’s ∨_{i}S_{i}^{2} → BF so that π_{2}(∨_{i}S_{i}^{2}) → (∨_{i}S_{i}^{2}) → π_{2}BF → π_{1}F has image B. Consider the classifying map X → BF of the given fibration with fundamental group data (∗). Using Van Kampen’s theorem it is easy to see that the fibration with classifying map X ∨ ∨_{i}S_{i}^{2} → BF has fundamental group data (∗∗). The total space of this fibration is the union of the total spaces of X → BF and ∨_{i}S_{i}^{2} → BF along F, so we may use a Van Kampen type theorem of Siebenmann [10] to compute the finiteness obstruction of the total space, thus computing the transfer map for one fibration and hence by Theorem 5.2 for any fibration with fundamental group data (∗∗) and the result follows. □

Example 5.5. Let S^{1} → Y → X be a fibration such that the fundamental group data Z/nZ → π_{1}Y → π_{1}X, π_{1}Y → Z/2Z is reduction mod n of some exact sequence Z → ρ → π_{1}X with the action of ρ on Z given by ρ → π_{1}Y → Z/2Z. The transfer map associated to the fibration S^{1} = K(Z, 1) → K(ρ, 1) → K(π_{1}X, 1) is described algebraically in [9]. (A projective Zπ_{1}X-module has homological dimension 1 as Zρ module and may thus be considered an element of K̃_{0}(Zρ).) By the above corollary the finiteness obstruction of Y is determined as the image of the finiteness obstruction of X in the composition K̃_{0}(Zπ_{1}X) → K̃_{0}(Zρ) → K̃_{0}(Zπ_{1}Y). This gives a new proof of Ehrlich’s result that if the fundamental group sequence of an orientable S^{1}-fibration is pseudo abelian, then the finiteness obstruction of the total space is 0. This follows since one easily sees that a pseudo abelian extension is the reduction mod n of a pseudo abelian integral extension, and an easy calculation shows that the transfer map of a pseudo abelian integral extension is 0 [9].

Remark. The above example holds in the Wh case as well (using [8]).

6. An algebraic application

In view of the results of the preceding sections, it becomes of some interest to compute G_{1}F for some more spaces. In the original paper [4], Gottlieb proves that G_{1}F = Cπ_{1}F for H-spaces and K(π, 1)’s. He also proves that if χ(F) ≠ 0 then G_{1}(F) = 0. In this section we will compute G_{1}(X) for X a (π, n) polarized space, i. e. a finitely dominated CW-complex X with an isomorphism π_{1}(X) ≅ π and a homotopy equivalence X̃ ≃ S^{n−1} (we assume n even and π acting orientation preserving on X̃). Let k(X) ∈ H^{n}(K(π, 1); Z) be the first k-invariant of X. It is proved in [6] that π_{0}H(X, ∗) → Aut π and π_{0}H(X) → Out(π) are monic and the image are those automorphisms of π that preserve this k-invariant.

Lemma 6.1. Let X be a polarized space as above. Then G_{1}(X) = Cπ.

Proof. Consider the commutative diagram of exact sequences

\xymatrix{1 \ar[r] & π_{1}X/G_{1}X \ar[d] \ar[r] & π_{0}(H(X, ∗)) \ar[d] \ar[r] & π_{0}(H(X)) \ar[d] \ar[r] & 1 \\ 1 \ar[r] & π_{1}X/Cπ \ar[r] & Aut π \ar[r] & Out π \ar[r] & 1}

Since π_{0}(H(X, ∗)) → Aut π is monic it follows that G_{1}(X) = Cπ. □

Corollary 6.2. Let X be a polarized space with fundamental group π and k-invariant k(X) ∈ H^{n}(K(π, 1); Z). Let 1 → π → ρξ → 1 be an exact sequence and assume that the action of ξ on π preserves k(X). The there exists a fibration X → E → B with fundamental group sequence isomorphic to the above exact sequence.

Proof. By assumption, the homomorphism ξ → Out(π) factors through π_{0}H(X). The obstruction to the existence of a fibration of Proposition 3.2 must be 0 since it is equal to the obstruction to the existence of the given exact sequence. □

Corollary 6.3. Let 1 → π → p → ξ → 1 be an exact sequence of finitely presented groups, π a group with periodic cohomology of period n (n even). Assume that the action of ξ on π preserves some generator of H^{n}(K(π, 1); Z). Then any projective Z[ξ]-modules M admits a resolution of finitely generated projectives over Z[ρ] which is periodic of period M.

Proof. Let k be some generator of H^{n}(K(π, 1); Z) which ξ preserves. We may then [6] find a polarized space with k as k-invariant. The result now follows from Corollary 6.2 by considerations analogous to Corollary 7.2 of [9] since Corollary 6.2 provides the relevant realizability result. □

References

[1] G. Allaud, On the classification of fiber spaces, Math. Z. 92 (1966), 110–125.

[2] D. R. Anderson, The Whitehead torsion of a fiber homotopy equivalence, Michigan Math. J. 21 (1974), 171–180.

[3] K. Ehrlich, Fibrations and a transfer map in algebraice K-theory, J. Pure Appl. Algebra 14 (1979), 131–136.

[4] D. Gottlieb, A certain subgroup of the fundamental group, Amer. J. Math. 87 (19688), 840–856.

[5] S. MacLane, Homotopy, Springer, 1963.

[6] I. Madsen, Smooth spherical space forms, Geometric Applications of Homotopy Theory (Evanston Ill, 1977), Lecture Notes in Mathematics, vol. 657, Springer, Berlin, 1978, pp. 303–352.

[7] P. May, Semi Simplicial Objects in Algebraic Topology, Van Nostrand, New York, 1967.

[8] H. J. Munkholm, Whitehead torison for PL fiber homotopy equivalences, Algebraic topology, (Waterloo, 1978), Lecture Notes in Mathematics, vol. 741, Springer, Berlin, 1979, pp. 90–101.

[9] H. J. Munkholm and E. K. Pedersen, On the Wall finiteness obstruction for some fibrations, Trans. Amer. Math. Soc. 261 (1980), 529–545.

[10] L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than 5, Ph.D. thesis, Princeton University, 1965.

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

University of Odense, 5230 Odense M, Denmark