“The Universal Functorial Lefschetz Invariant”

by

Wolfgang Lück

Abstract

We introduce the universal functorial Lefschetz invariant for endomorphisms of finite CW -complexes in terms of Grothendieck groups of endomorphisms of finitely generated free modules. It encompasses invariants like Lefschetz number, its generalization to the Lefschetz invariant, Nielsen number and L^{2}-torsion of mapping tori. We examine its behaviour under fibrations.

Key words: Universal functorial Lefschetz invariants, Grothendieck group of endomorphisms of modules, transfer maps

AMS-classification number: 57Q99, 19A99

Introduction

Given an endomorphism f : X ⟶ X of finite CW -complex X (with π_{0}(f) = id), we introduce an abelian group U(f) and an invariant u(f) in 2.2 based on an algebraic invariant for chain complexes which we will define in Definition 1.2. The algebraic version of U(f) for a ring R with an endomorphism ϕ : R ⟶ R is the Grothendieck group of ϕ-linear endomorphisms of finitely generated free R-modules. The pair (U, u) is a functorial Lefschetz invariant on the category End(C) of endomorphisms f : X ⟶ X of finite CW -complexes (with π_{0}(f) = id) in the sense of Definition 2.3, i.e. U is a functor from End(C) into the category of abelian groups and for any object f : X ⟶ X there is an invariant u(f) ∈ U(f) such that (U, u) satisfies a push out formula, for a morphism h : (X, f) ⟶ (Y, g) in End(C) U(h) : U(f) ⟶ U(g) depends only on the homotopy class of h, u(id : ∅ ⟶ ∅) = 0 and U(h) maps u(f) to u(g) and is bijective if h is a homotopy equivalence. We call a functorial Lefschetz invariant (A, a) universal if for any other functorial Lefschetz invariant (B, b) there is precisely one natural transformation ξ : A ⟶ B which satisfies ξ(f)(a(f)) = b(f) for all objects f : X ⟶ X in End(C) (see Definition 2.4). One of the main results of the paper is proven in Section 4

Theorem 2.5. The pair (U, u) is the universal functorial Lefschetz invariant for endomorphisms of finite CW -complexes.

The universal functorial Lefschetz invariant is unique and carries maximal information compared with any other functorial Lefschetz invariant. In particular any result about the universal functorial Lefschetz invariant carries over to any other functorial Lefschetz invariant. We will give in Section 3 examples of functorial Lefschetz invariants such as the (classical) Lefschetz number, its generalization to the Lefschetz invariant and the Nielsen number which have been extensively studied in the literature, and of a new one which is essentially given by the L^{2}-torsion of the mapping torus. The last one can be used to compute the volume of a hyperbolic closed 3-manifold given by a mapping torus of a pseudo-Anosov selfhomeomorphism f of a closed hyperbolic 2-dimensional manifold and hence the volume can be derived from u(f). We do not know how to get the volume from the other functorial Lefschetz invariants mentioned above.

In Section 5 we investigate the behaviour of u(f) under fibrations by assigning to a fibration a transfer map 5.6 which computes the invariants on the total space level by the one on the basis level (Theorem 5.8). We investigate this transfer map algebraically in Section 6. We obtain a down-up-formula (see Lemma 6.5), give explicit calculations for S^{n} as fiber (see Example 6.9) and prove

Theorem 6.7. Let f : X ⟶ X be a S^{1}-endomorphism of a finite S^{1}-CW -complex X. Denote by i : (X^{S^{1}}, f^{S^{1}}) ⟶ (X, f) the morphism in End(C) induced by the inclusion of the fixed point set. Then we have:

U(i)(u(X^{S^{1}}, f^{S^{1}})) = u(X, f) ∈ U(X, f).

In particular u(X, f) vanishes if the S^{1}-action has no fixed points.

In Section 7 we construct a diagram

7.1 \begin{matrix}U(R, ϕ) & ⟶^{τ} & U(R̂[t, t^{−1}]ϕ̂, id) \\ ^{η}↓ & & ^{η}↓ \\ ∏_{m≥1} Λ(R, ϕ^{m}) & \underrightarrow{τ=∏_{m≥1} τ_{m}} & ∏_{m≥1} Λ(R̂[t, t^{−1}]ϕ̂, id) \end{matrix}

The horizontal maps are given by the mapping torus construction whereas the vertical maps are given by the Lefschetz invariants of the various iterates of an endomorphism. In the case where R is commutative and ϕ = id this corresponds to the passage to the Lefschetz Zeta-function. We discuss the question which of the maps are injective.

The paper is organized as follows:

0. Introduction
1. The universal Lefschetz invariant for chain complexes
2. The universal Lefschetz invariant for CW -complexes
3 Examples
4. Proof of the universal property
5. The construction of the transfer map
6. Properties of the transfer map
7. The mapping torus approach

References

Finally the author wants to thank the referee for its very detailed and very helpful report and for pointing out an error in Section 7 in the first version.

1 The universal Lefschetz invariant for chain complexes

In this section we introduce the universal Lefschetz invariant for finite free chain complexes. This is the algebraic version of the universal Lefschetz invariant for spaces which we will introduce in Definition 2.4. Modules are always left-modules unless explicitely stated differently.

Recall that an R-chain complex is finitely generated resp. free if each of its chain modules have this property. It is called finite-dimensional if C_{n} is zero for n < 0 and n > N for some N. We call it finite if it is both finitely generated and finite-dimensional. Given an R-module F, let el(F, n) be the associated n-dimensional elementary chain complex which is concentrated in dimension n and n − 1 and has as n-th differential id : F ⟶ F.

Definition 1.1. Let R be a an associative ring with unit and ϕ : R ⟶ R be a ring homomorphism respecting the unit. An additive invariant for the category of ϕ-endomorphisms of finite free R-chain complexes is a pair (A, a) which consists of an abelian group A and a function which assigns to each R-chain map f : C ⟶ ϕ^{∗}C for C a finite free R-chain complex an element

a(f) ∈ A

where ϕ^{∗}C_{∗} is the R-chain complex obtained from C_{∗} by restriction with ϕ. such that the following holds:

1. Additivity

For a commutative diagram of finite free R-chain complexes with exact rows

0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0
f↓  ^{g}↓  ^{g}↓
0 ⟶ ϕ^{∗}C ⟶^{ϕ^{∗}i} ϕ^{∗}D ⟶^{ϕ^{∗}p} ϕ^{∗}E ⟶ 0

we have

a(f) − a(g) + a(h) = 0;

2. Homotopy invariance

Let f, g : C ⟶ ϕ^{∗}C be R-chain maps of finite free R-chain complexes. If f and g are R-chain homotopic, then

a(f) = a(g);

3. Elementary chain complexes

We have for any finitely generated free R-module F and n ≥ 1

a(0 : el(F, n) ⟶ ϕ^{∗} el(F, n)) = 0.

We call an additive invariant (U, u) universal if for any additive invariant (A, a) there is precisely one homomorphism ξ : U ⟶ A of abelian groups satisfying ξ(u(f)) = a(f) for all f : C ⟶ ϕ^{∗}C.

In particular an R-map f : F ⟶ ϕ^{∗}Q is map of abelian groups such thatf(rx) = ϕ(r)f(x) holds for all r ∈ R and x ∈ F.

Definition 1.2. Let R be a ring and ϕ : R ⟶ R be a ring homomorphism. Let U(R, ϕ) be the abelian group defined by generators and relations as follows. Generators [A] are given by (n, n)-matrices A with entries in R for n ≥ 1. If A is in block form for square matrices B and D

A = ( \begin{matrix}B & C \\ 0 & D \end{matrix} ) ,

then [A] = [B] + [D]. If A is a (n, n)-matrix, U is an invertible (n, n)-matrix and ϕ(U) denotes the matrix obtained from U by applying ϕ to each entry, then [ϕ(U)AU^{−1}] = [A].

Given a R-map f : F ⟶ ϕ^{∗}F for a finitely generated free R-module F, define

[f] ∈ U(R, ϕ)

by [A], where A is the matrix descibing ϕ^{∗}u^{−1} ∘ f ∘ u : R^{n} ⟶ ϕ^{∗}R^{n} for any R-isomorphism u : R^{n} ⟶ F. Given an R-chain map f : C ⟶ ϕ^{∗}C for C a finite free R-chain complex C, we define:

u(f) := ∑_{n≥0}(−1)^{n} · [f_{n} : C_{n} ⟶ ϕ^{∗}C_{n}] ∈ U(R, ϕ).

We have defined U(R, ϕ) in terms of matrices since the square matrices form a set, whereas the R-maps F ⟶ ϕ^{∗}F for finitely generated free R-modules do not form a set. Notice that for each commutative diagram of R-maps of finitely generated free R-modules with exact rows

0 ⟶ F_{1} ⟶^{i} F_{2} ⟶^{p} F_{3} ⟶ 0
f_{1}↓  ^{f_{2}}↓  ^{f_{3}}↓
0 ⟶ ϕ^{∗}F_{1} ⟶^{ϕ^{∗}i} ϕ^{∗}F_{2} ⟶^{ϕ^{∗}p} ϕ^{∗}F_{3} ⟶ 0

we get the relation

[f_{1}] − [f_{2}] + [f_{3}] = 0.

Given rings with endomorphisms (R, ϕ) and (S, ψ) and a ring homomorphism h : R ⟶ S with ψ ∘ h = h ∘ ϕ, induction with h induces a homomorphism

h_{∗} : U(R, ϕ) ⟶ U(S, ψ)  [g : F ⟶ ϕ^{∗}F] ↦ [h_{∗}g : h_{∗}F ⟶ ψ^{∗}h_{∗}F],  (1.3)

where h_{∗}g sends s ⊗ f ∈ S ⊗_{h} F to ψ(s) ⊗ g(f) ∈ ψ^{∗}(S ⊗_{h} F).

Theorem 1.4. (U(R, ϕ), u) is the universal additive invariant for ϕ-endomorphisms of finite free R-chain complexes.

Proof. We first show that (U(R, ϕ), u) is an additive invariant. Additivity follows directly from the definitions of u and U(R, ϕ). Obviously u(0 : el(F, n) ⟶ el(F, n)) = 0. It remains to check homotopy invariance. Let h : C ⟶ ϕ^{∗}C be an R-chain homotopy from f to g. Denote by ΣC the suspension of C and by cone(C) the mapping cone of C. Then one obtains an R-chain map k : cone(C) ⟶ cone(C) by putting

k_{n} = ( \begin{matrix}f_{n−1} & 0 \\ h_{n−1} & g_{n} \end{matrix} ) : C_{n−1} ⊕ C_{n} ⟶ C_{n−1} ⊕ C_{n}

such that there is a commutative diagram of R-chain complexes with exact rows:

0 ⟶ C ⟶^{i} cone(C) ⟶^{p} ΣC ⟶ 0
f↓  ^{k}↓  ^{Σg}↓
0 ⟶ ϕ^{∗}C ⟶^{ϕ^{∗}i} ϕ^{∗} cone(C) ⟶^{ϕ^{∗}p} ϕ^{∗}ΣC ⟶ 0

We conclude from additivity:

u(f) − u(g) = u(f) + u(Σg) = u(k).

Notice that cone(C) is a contractible R-chain complex. Hence it suffices to show

u(f : C ⟶ ϕ^{∗}C) = 0,

provided C is contractible. We do this by induction over the dimension of C. If d ≤ 2, then the claim follows from the definitions of u and U(R, ϕ). The induction step from d ≥ 2 to d + 1 is done as follows. Let D be the R-subchain complex of C given by D_{d+1} = C_{d+1}, D_{d} = ker(c_{d}) and D_{p} = 0 for p ≠ d, d + 1. Let E be the cokernel of the inclusion D ⟶^{i} C. Since C is a finite free contractible R-chain complex, we can assume without loss of generality that D and E are finite free contractible R-chain complexes, otherwise add to C the elementary chain complex el(C_{d+1}, d). The R-chain map f induces R-chain maps g : D ⟶ ϕ^{∗}D and h : E ⟶ ϕ^{∗}E. Now one gets from additivity:

u(g) − u(f) + u(h) = 0.

We get from the induction beginning u(g) = u(h) = 0. We conclude u(f) = 0. This finishes the proof that (U(R, ϕ), u) is an additive invariant.

It remains to check the universal property. Let (A, a) be an additive invariant for ϕ-endomorphisms of finite free chain complexes. There is precisely one homomorphism

ξ : U(R, ϕ) ⟶ A

which sends a generator represented by an R-map f : F ⟶ ϕ^{∗}F to a(f), where we interpret f as an R-chain map of finite 0-dimensional R-chain complexes. It remains to show for an R-chain map f : C ⟶ ϕ^{∗}C for C a finite free R-chain complex

a(f) = ∑_{n≥0}(−1)^{n} · ξ(u(f_{n})).

We do this by induction over the dimension d of C. The induction beginning d = 0 follows from the definition of ξ. Let D be the R-chain complex which is concentrated in dimension d and satisfies D_{d} = C_{d}. Denote by C|_{d−1} the d − 1-dimensional R-chain complex obtained by truncating C. Let g : C|_{d−1} ⟶ ϕ^{∗}C|_{d−1} and h : D ⟶ ϕ^{∗}D be the R-chain maps induced by f. From additivity and the obvious exact sequence of R-chain complexes 0 → C|_{d−1} ⟶C ⟶D → 0 we conclude

a(f) = a(g) + a(h).

Because of the induction hypothesis applied to C|_{d−1} and Σ^{−1}D it suffices to show

a(h) = −a(Σ^{−1}h).

There is an obvious exact sequence 0 → Σ^{−1}D ⟶el(C_{d}, d) ⟶D → 0 and an R-chain map \overline{h} : el(C_{d}, d) ⟶ ϕ^{∗} el(C_{d}, d) compatible with the exact sequence above. We conclude

a(Σ^{−1}h) + a(h) = a(\overline{h}) = a(0 : el(C_{d}, d) ⟶ ϕ^{∗} el(C_{d}, d)) = 0.

This finishes the proof of Theorem 1.4.

There is a canonical homomorphism

s : ℤ ⟶ U(R, ϕ) m − n ↦ [0 : R^{m} ⟶ ϕ^{∗}R^{m}] − [0 : R^{n} ⟶ ϕ^{∗}R^{n}].

Suppose for simplicity that R has the property that R^{n} ≅ R^{m} implies n = m. This condition is satisfied in our main example, namely in the case where R is the integral group ring of a group. Then each finitely generated R-module F has a well-defined dimension dim_{R}(F) ∈ ℤ and we obtain a homomorphism

dim : U(R, ϕ) ⟶ ℤ  [F, f] ↦ dim_{R}(F).  (1.5)

satisfying dim ∘s = id. Recall for a finite free R-chain complex C that its Euler characteristic is defined by

χ(C) := ∑_{n≥0}(−1)^{n} · dim_{R}(C) ∈ ℤ.

Now we can show that a crucial property of Lefschetz type or trace type invariants, commutativity, follows from additivity and homotopy invariance.

Lemma 1.6. Let v : C ⟶ D and f : D ⟶ ϕ^{∗}C be R-chain maps of finite free R-chain complexes. Then we get

u(f ∘ v) + s(χ(D)) = u(ϕ^{∗}v ∘ f) + s(χ(C)).

Proof. Consider the following commutative diagram

D ⊕ C \underrightarrow{( \begin{matrix}0 & 0 \\ f & f ∘ v \end{matrix} )} ϕ^{∗}D ⊕ ϕ^{∗}C
( \begin{matrix}1 & v \\ 0 & 1 \end{matrix} )↓  ↓( \begin{matrix}1 & ϕ^{∗}v \\ 0 & 1 \end{matrix} )
D ⊕ C \overrightarrow{( \begin{matrix}ϕ^{∗}v ∘ f & 0 \\ f & 0 \end{matrix} )} ϕ^{∗}D ⊕ ϕ^{∗}C

Since the vertical arrows are isomorphisms, we get from additivity

u ( \begin{matrix}0 & 0 \\ f & f ∘ v \end{matrix} ) = u ( \begin{matrix}ϕ^{∗}v ∘ f & 0 \\ f & 0 \end{matrix} ) .

We derive from additivity

u(f ∘ v) + u(0 : D ⟶ ϕ^{∗}D) = u ( \begin{matrix}0 & 0 \\ f & f ∘ v \end{matrix} );
u(ϕ^{∗}v ∘ f) + u(0 : C ⟶ ϕ^{∗}C) = u ( \begin{matrix}ϕ^{∗}v ∘ f & 0 \\ f & 0 \end{matrix} ) .

This finishes the proof of Lemma 1.6.

Example 1.7. Suppose that R is commutative and that ϕ = id. Then U(R, id) is computed in [1, page 377] and [16, Corollary 3, page 442]. Namely, it is given by

U(R, id) ⟶ ℤ × \{ \frac{1 + a_{1}t + … a_{n}t^{n}}{1 + b_{1}t + … b_{m}t^{m}} | a_{i}, b_{i} ∈ R\} [F, f] ↦ (dim_{R}(F), det(1 − tf)),

where det(1 − tf) is the characteristic polynomial in the variable t of f.

Let c : ℂ ⟶ ℂ be complex conjugation. As an illustration we want to investigate U(ℂ, c). For a complex number z ∈ ℂ define ℂ-linear maps

R_{z} : ℂ ⟶ c^{∗}ℂ  u ↦ zc(u);
S_{z} : ℂ ⊕ ℂ ⟶ c^{∗}(ℂ ⊕ ℂ)  (a, b) ↦ (zc(b), c(a)).

Recall that we have introduced the map dim in 1.5. Define a map

η : U(ℂ, c) ⟶ ∏_{n≥1} ℂ  [f] ↦ (tr_{ℂ}(f^{2n}))_{n≥1} .  (1.8)

Theorem 1.9. 1. U(ℂ, c) is the free abelian group with basis

B := {[R_{r}] | r ∈ ℝ, r ≥ 0} ∐{[S_{s}] | s ∈ ℝ, s < 0} ∐{[S_{z}] | z ∈ ℂ, ℑ(z) > 0};

2. The map dim ×η : U(ℂ, c) ⟶ ℤ × ∏_{n≥1} ℂ is injective;

Proof. We first show that the set

{[R_{z}] | z ∈ ℂ} ∐ {[S_{z}] | z ∈ ℂ}  (1.10)

generates U(ℂ, c). We show by induction over the dimension of the complex vector space V that the class [f] ∈ U(ℂ, c) of a ℂ-linear map f : V ⟶ c^{∗}V is a linear combination of elements of this set. Notice that f^{2} : V ⟶ V is a ℂ-linear endomorphism and hence has a non-trivial eigenvector v ∈ V with eigenvalue μ. Consider the subspace U of V spanned by v and f(v). Obviously f induces ℂ-linear maps f_{0} : U ⟶ c^{∗}U and f_{1} : V/U ⟶ c^{∗}V/U such that we get in U(ℂ, c)

[f] = [f_{0}] + [f_{1}].

If U is different from V, the induction step follows from the induction hypothesis. Hence it remains to treat the case U = V. Suppose that v and f(v) are linearly dependent. Then there is z ∈ ℂ with f(v) = zv and hence [f] = [R_{z}]. Hence it remains to treat the case where {v, f(v)} is a basis for V. If we conjugate f with the ℂ-isomorphism ℂ^{2} ⟶ V which maps (1, 0) to v and (0, 1) to f(v), then we obtain S_{μ} and hence [f] = [S_{μ}].

Next we want to show that the set B defined in Theorem 1.9.1 generates U(ℂ, c). For that purpose it suffices to verify in U(ℂ, c) the relations

[S_{r^{2}}] = 2[R_{r}]  for r ∈ ℝ^{>0};  (1.11)
[R_{|z|}] = [R_{z}]  for z ∈ ℂ;  (1.12)
[S_{z}] = [S_{c(z)}]  for z ∈ ℂ.  (1.13)

We get the following equations of maps from ℂ ⊕ ℂ to ℂ ⊕ ℂ

( \begin{matrix}r & 1 \\ 0 & i \end{matrix} ) · ( \begin{matrix}0 & c \\ R_{r^{2}} & 0 \end{matrix} ) = ( \begin{matrix}R_{r^{2}} & R_{r} \\ R_{ir^{2}} & 0 \end{matrix} )
= ( \begin{matrix}R_{r} & 0 \\ R_{ir} & R_{r} \end{matrix} ) · ( \begin{matrix}r & 1 \\ 0 & i \end{matrix} ) .

Now 1.11 follows from

[S_{r^{2}}] = [( \begin{matrix}R_{r} & 0 \\ R_{ir} & R_{r} \end{matrix} )]
= [R_{r}] + [R_{r}].

For z ∈ ℂ with z ≠ 0 choose ω ∈ ℂ satisfying ω^{−1}zc(ω) = |z|. If we conjugate R_{z} with ω · id, we obtain R_{|z|} and hence 1.12 follows.

We obtain 1.13 from

( \begin{matrix}0 & 1 \\ c(z) & 0 \end{matrix} ) · ( \begin{matrix}0 & c \\ R_{z} & 0 \end{matrix} ) = ( \begin{matrix}R_{z} & 0 \\ 0 & R_{c(z)} \end{matrix} )
= ( \begin{matrix}0 & c \\ R_{c(z)} & 0 \end{matrix} ) · ( \begin{matrix}0 & 1 \\ c(z) & 0 \end{matrix} )

Since the set defined in 1.10 generates U(ℂ, c), we conclude from 1.11, 1.12 and 1.13 that the set B generates U(ℂ, c).

Finally we show that the set (dim ×η) (B) is a ℤ-linear independent subset of ℤ × ∏_{n≥1} ℂ. Notice that then Theorem 1.9 follows since B generates U(ℂ, c). Notice that [R_{0}] lies in the kernel of η and is mapped to 1 under dim. Hence it suffices to show for sequences 0 < r_{1} < r_{2} < … r_{d} and 0 > s_{1} > s_{2} > … > s_{e} of real numbers, and a sequence z_{1}, z_{2}, …, z_{f} of pairwise distinct complex numbers with positive imaginary parts that for any sequences of integers λ_{1}, …, λ_{d}, μ_{1}, …, μ_{e} and ν_{1}, …, ν_{f} which satisfy

∑_{i=1}^{d} λ_{i} · η([R_{r_{i}}]) + ∑_{j=1}^{e} μ_{j} · η([S_{s_{j}}]) + ∑_{k=1}^{f} μ_{k} · η([S_{z_{k}}]) = 0  (1.14)

the equations

λ_{i} = 0  for i = 1, … d;  (1.15)
μ_{j} = 0  for j = 1, … e;  (1.16)
ν_{k} = 0  for k = 1, … f.  (1.17)

hold. One easily computes

η([R_{r_{i}}] = (r_{i}^{2n})_{n≥1};
η([S_{s_{j}}]) = (2s_{j}^{n})_{n≥1};
η([S_{z_{k}}]) = (z_{k}^{n} + c(z_{k})^{n})_{n≥1}.

Hence we get from 1.14

∑_{i=1}^{d} λ_{i}r_{i}^{2n} + ∑_{j=1}^{e} μ_{j}2s_{j}^{n} + ∑_{k=1}^{f} ν_{k}(z_{k}^{n} + c(z_{k})^{n}) = 0  for n ≥ 1.  (1.18)

If we apply the next Lemma 1.19 to the sequences

w_{n} := r_{1}^{2}, … r_{d}^{2}, s_{1}, … s_{e}, z_{1}, c(z_{1}), … z_{f}, c(z_{f});
v_{n} := λ_{1}, … λ_{d}, 2μ_{1}, …, 2μ_{e}, ν_{1}, ν_{1}, … ν_{f}, ν_{f};

then we get 1.15, 1.16 and 1.17. This finishes the proof of Theorem 1.9 except for the proof of Lemma 1.19 we will give next.

Lemma 1.19. Let w_{1}, …, w_{l} be a sequence of pairwise distinct non-zero complex numbers and v_{1}, …, v_{l} be a sequence of complex numbers satisfying

∑_{i=1}^{l} v_{i}w_{i}^{n} = 0 for n = 1, 2, …, l.

Then v_{i} = 0 for i = 1, 2 … l.

Proof. We have to show that the l elements (w_{1}^{n}, w_{2}^{n}, … w_{l}^{n}) ∈ ℂ^{l} for n = 1, 2 … l are ℂlinearly independent. This follows from the following determinant computation involving Vandermonde’s determinant

| \begin{matrix}w_{1} & w_{2} & … & w_{l} \\ w_{1}^{2} & w_{2}^{2} & … & w_{l}^{2} \\ ⋮ & ⋮ & ... & ⋮ \\ w_{1}^{l} & w_{2}^{l} & … & w_{l}^{l} \end{matrix} | = w_{1}w_{2} … w_{l} | \begin{matrix}1 & 1 & … & 1 \\ w_{1} & w_{2} & … & w_{l} \\ ⋮ & ⋮ & ... & ⋮ \\ w_{1}^{l−1} & w_{2}^{l−1} & … & w_{l}^{l−1} \end{matrix} |

= w_{1}w_{2} … w_{l} ∏_{i>j}(w_{i} − w_{j})
≠ 0.

This finishes the proof of Lemma 1.19 and Theorem 1.9.

2 The universal Lefschetz invariant for CW -complexes

In this section we want to apply the chain complex invariants of Section 1 to endomorphisms f : X ⟶ X of finite CW -complexes satisfying π_{0}(f) = id. Suppose for the moment that X is connected. We make the following choices, namely, of a point x ∈ X and a path w in X from y = fx to x. Next we want to define an abelian group U(f, x, w) and an invariant

u(f, x, w) ∈ U(f, x, w).  (2.1)

For this purpose we make additional choices, namely of a model of the universal covering p : X̃ ⟶ X and a point x̃ ∈ X̃ satisfying p(x̃) = x. Let ỹ ∈ X̃ be the point satisfying p(ỹ) = y such that w lifts to path in X̃ from ỹ to x̃. There is precisely one lift f̃: X̃ ⟶ X̃ satisfying f̃(x̃) = ỹ. There is a specific left-action of π_{1}(X, x) on X̃ uniquely determined by the choice of x̃. Namely, for z̃ ∈ X̃ and u ∈ π_{1}(X, x) let uz̃ ∈ X̃ be the point such that p(uz̃) = p(z̃) holds and for any paths ã from x̃ to z̃ and b̃ from x̃ to uz̃ the loop p(b̃) ∗ p(ã^{−}) represents u ∈ π_{1}(X, x). Let c_{w} : π_{1}(X, y) ⟶ π_{1}(X, x) be the homomorphism sending u to w^{−} ∗ u ∗ w. Let ϕ = ϕ(f, x, w) : π_{1}(X, x) ⟶ π_{1}(X, x) be the composition c_{w} ∘ π_{1}(f, x). Then f̃: X̃ ⟶ X̃ is ϕ-equivariant. The ring homomorphism ℤπ_{1}(X, x) ⟶ ℤπ_{1}(X, x) induced by the group homomorphism ϕ is also denoted by ϕ. We get a ℤπ_{1}(X, x)-chain map C(f̃) : C(X̃) ⟶ ϕ^{∗}C(X̃). Define (see Definition 1.2)

U(f, w, x) := U(ℤπ_{1}(X, x), ϕ(f, w, x))

and

u(f, x, w) := u(C(f̃)) ∈ U(f, w, x).

We have to verify that u(f, w, x) is independent of the choice of X̃ and x̃ ∈ p^{−1}(x). Let X̃′ and x̃′ be different choices. The identity map id : (X, x) ⟶ (X, x) lifts uniquely to a π_{1}(X, x)-equivariant homeomorphism ĩd : (X̃, x̃) ⟶ (X̃′, x̃′) such that f̃′ ∘ ĩd = ϕ^{∗}ĩd ∘ f̃ holds. Theorem 1.4 and Lemma 1.6 imply

u(C(f̃)) = u(C(f̃′)) ∈ U(f, x, w).

Hence u(f, x, w) depends only on (x, w).

Next we examine the dependency on (x, w). Let (x_{k}, w_{k}) for k = 0, 1 be two such choices. We want to construct a homomorphism

μ = μ(x_{0}, w_{0}, x_{1}, w_{1}) : U(f, x_{0}, w_{0}) ⟶ U(f, x_{1}, w_{1}).

To do this we choose a path v from x_{0} to x_{1}. Recall that for a path a we get by conjugation a homomorphism c_{a} : π_{1}(X, a(0)) ⟶ π_{1}(X, a(1)). One easily checks

ϕ_{1} ∘ c_{v} = c_{w_{0}^{−}∗f(v)∗w_{1}} ∘ Φ_{0},

where ϕ_{k} : π_{1}(X, x_{k}) ⟶ π_{1}(X, x_{k}) is the endomorphism ϕ with respect to the choice (x_{k}, w_{k}) for k = 0, 1. Given a finitely generated free ℤπ_{1}(X, x_{0})-module F, define an isomorphism of ℤπ_{1}(X, x_{1})-modules

ρ(F) : (c_{v})_{∗}ϕ_{0}^{∗}F ⟶ ϕ_{1}^{∗}(c_{v})_{∗}F

by

ℤπ_{1}(X, x_{1}) ⊗_{c_{v}} ϕ_{0}^{∗}F ⟶ ϕ_{1}^{∗} (ℤπ_{1}(X, x_{1}) ⊗_{c_{v}} F) g ⊗ f ↦ Φ_{1}(g) ⊗ u^{−1}f,

where u ∈ π_{1}(X, x_{0}) is given by w_{0} ∗ f(v) ∗ w_{1} ∗ v^{−}. In order to check that this is well-defined we must show that gc_{v}(h) ⊗ f and g ⊗ ϕ_{0}(h)f are mapped to the same elements, i.e. we must show in ℤπ_{1}(X, x_{1}) ⊗_{c_{v}} F

ϕ_{1}(gc_{v}(h)) ⊗ u^{−1}f = ϕ_{1}(g) ⊗ u^{−1}ϕ_{0}(h)f.

We compute

ϕ_{1}(gc_{v}(h)) ⊗ u^{−1}f = ϕ_{1}(g)ϕ_{1}(c_{v}(h)) ⊗ u^{−1}f
= ϕ_{1}(g) ⊗ c_{v}^{−1} ∘ ϕ_{1} ∘ c_{v}(h)u^{−1}f
= ϕ_{1}(g) ⊗ c_{v}^{−1} ∘ c_{w_{0}^{−}∗f(v)∗w_{1}} ∘ ϕ_{0}(h)u^{−1}f
= ϕ_{1}(g) ⊗ c_{w_{0}^{−}∗f(v)∗w_{1}∗v^{−}} ∘ ϕ_{0}(h)u^{−1}f
= ϕ_{1}(g) ⊗ c_{u} ∘ ϕ_{0}(h)u^{−1}f
= ϕ_{1}(g) ⊗ u^{−1}ϕ_{0}(h)uu^{−1}f
= ϕ_{1}(g) ⊗ u^{−1}ϕ_{0}(h)f.

We define the desired homomorphism

μ = μ(x_{0}, w_{0}, x_{1}, w_{1}) : U(f, x_{0}, w_{0}) ⟶ U(f, x_{1}, w_{1})

by

μ ([f : F ⟶ ϕ_{0}^{∗}F]) = [(c_{v})_{∗}F ⟶^{(c_{v})_{∗}f} (c_{v})_{∗}ϕ_{0}^{∗}F ⟶^{ρ(F)} ϕ_{1}^{∗}(c_{v})_{∗}F].

One easily checks that this map is independent of the choice of v. Moreover, it sends u(f, x, w) to u(f, x_{1}, w_{1}) because of Theorem 1.4 since there is the following commutative diagram

X̃ \underrightarrow{f̃_{0}} X̃
id↓  ↓^{l_{u}}
X̃ \underrightarrow{f̃_{1}} X̃

One easily checks

μ(x_{0}, w_{0}, x_{2}, w_{2}) = μ(x_{1}, w_{1}, x_{2}, w_{2}) ∘ μ(x_{0}, w_{0}, x_{1}, w_{1});
μ(x_{0}, w_{0}, x_{0}, w_{0}) = id .

We define U(f) as the abelian group which is the set of equivalence classes of the equivalence relation on ∐_{(x,w)} U(f, x, w) generated by u ∼ c(x, x′, w, w′)(u) for u ∈ U(f, x, w). The collection of the u(f, x, w) determines an element

u(f) ∈ U(f).  (2.2)

The obvious map U(f, w, x) ⟶ U(f) is an isomorphism and sends u(f, x, w) to u(f) for all (x, w).

Recall that so far we have assumed that X is connected. If X has more than one path component, we will assume that f induces the identity on π_{0}(X) and we define U(f) by the direct sum over the path components C of X of the groups U(f|_{C}) and u(f) ∈ U(f) by the collection of the invariants u(f|_{C}).

Let C be the category having as objects finite CW -complexes and as morphisms maps between them. The category End(C) has as objects (X, f) endomorphisms f : X ⟶ X in C such that f induces the identity on π_{0}(X). A morphism h : (X, f) ⟶ (Y, g) in End(C) is a commutative square in C

\begin{matrix}X & ⟶^{f} & X \\ ^{h}↓ & & ↓^{h} \\ Y & ⟶^{g} & Y \end{matrix}

Given two such morphisms h_{i} : (X, f) ⟶ (Y, g) for i = 0, 1, a homotopy from h_{0} to h_{1} is given by a commutative square in C

X × [0, 1] ⟶^{f×id} X × [0, 1]
h↓  ↓^{h}
Y  ⟶^{g}  Y

such that the restriction of h to X × {i} agrees with h_{i} for i = 0, 1. If such a homotopy exists, we call h_{0} and h_{1} homotopic. A push out in End(C) is a commutative square in End(C)

(X_{0}, f_{0}) ⟶^{i_{1}} (X_{1}, f_{1})
i_{2}↓  ↓^{j_{1}}
(X_{2}, f_{2}) ⟶^{j_{2}} (X, f)

such that the commutative square in C

X_{0} ⟶^{i_{1}} X_{1}
i_{2}↓  ↓^{j_{1}}
X_{2} ⟶^{j_{2}} X

is a push out, f is the push out of f_{0}, f_{1} and f_{2}, i_{2} is an inclusion of CW -complexes, i_{1} is cellular and X has the CW -structure induced by the ones on X_{i} for i = 0, 1, 2.

Definition 2.3. A functorial Lefschetz invariant on the category of finite CW -complexes is a pair (Θ, θ) consisting of a functor

Θ : End(C) ⟶ ABEL

into the category of abelian groups and a function θ which assigns to any object (X, f) in End(C) an element

θ(X, f) ∈ Θ(X, f)

such that the following conditions are satisfied:

1. Additivity

For a push out in End(C)

(X_{0}, f_{0}) ⟶^{i_{1}} (X_{1}, f_{1})
i_{2}↓  ↓^{j_{1}}
(X_{2}, f_{2}) ⟶^{j_{2}} (X, f)

we get in Θ(X, f):

θ(X, f) = Θ(j_{1})(θ(X_{1}, f_{1})) + Θ(j_{2})(θ(X_{2}, f_{2})) − Θ(j_{0})(θ(X_{0}, f_{0})),

where j_{0} is j_{1} ∘ i_{1} = j_{2} ∘ i_{2};

2. Homotopy invariance

If h_{i} : (X, f) ⟶ (Y, g) are homotopic morphisms in End(C) for i = 0, 1, then:

Θ(h_{0}) = Θ(h_{1});

3. Invariance under homotopy equivalence

If h : (X, f) ⟶ (Y, g) is a morphism in End(C) such that h : X ⟶ Y is a homotopy equivalence, then

Θ(h) : Θ(X, f) ⟶ Θ(Y, g)

is bijective and sends θ(X, f) to θ(Y, g);

4. Value at the empty set

θ(id : ∅ ⟶ ∅) = 0 ∈ Θ(∅, id).

Definition 2.4. We call a functorial Lefschetz invariant (U, u) universal if for any functorial Lefschetz invariant (Θ, θ) there is precisely one natural transformation τ : U ⟶ Θ such that τ(f) : U(f) ⟶ Θ(f) sends u(f) to θ(f) for any object f : X ⟶ X in End(C).

The following theorem is one of the main results of this paper. It explains why the invariant u encompasses a lot of known Lefschetz type invariants and other invariants as we will analyse in Section 3. We will give its proof in Section 4. Analogous results for finiteness obstructions and Whitehead torsion have been proven in [25], [28, Section 6]. Notice that U becomes a functor U : End(C) ⟶ ABEL by 1.3.

Theorem 2.5. The pair (U, u) defined in 2.2 is the universal functorial Lefschetz invariant for endomorphisms of finite CW -complexes in the sense of Definition 2.3.

For related universal properties of Lefschetz-type invariants for spaces with group actions we refer for instance to [22], [39].

Remark 2.6. In the definition of U(X, f) we could work with finitely generated projective modules instead of finitely generated free modules. This corresponds in geometry to the passage from finite CW -complexes to finitely dominated CW -complexes. Then the new group would be the direct sum of the version discussed here with

⊕_{C∈π_{0}(X)}K̃_{0}(ℤπ_{1}(C))

and the new invariant would be the sum with the old one and the collection of Wall’s finiteness obstructions o(C) ∈ K̃_{0}(ℤπ_{1}(C)) of the components C.

Remark 2.7. Let f_{k} : X ⟶ X for k = 0, 1 be homotopic endomorphisms of a finite CW - complex X with π_{0}(f) = id. Let h : f_{0} ≃ f_{1} be such a homotopy. Let (Θ, θ) be a functorial Lefschetz invariant. We obtain an isomorphism

Θ_{h} : Θ(X, f_{0}) ⟶^{Θ(k_{0})} Θ(X × [0, 1], h × pr_{[0,1]}) ⟶^{Θ(k_{1})^{−1}} Θ(X, f_{1}),  (2.8)

where k_{n} : X ⟶ X × [0, 1] maps x to (x, n) for n = 0, 1. Because of invariance under homotopy equivalence, Θ(k_{n}) is bijective for n = 0, 1 and

Θ_{h}(θ(X, f_{0})) = θ(X, f_{1}).

Notice, however, that Θ_{h} does depend on h, or more precisely, on the homotopy class of the path h(x, −) as the following concrete calculation for the universal invariant shows.

Suppose for simplicity in the sequel that X is connected. Let x be a base point and w_{k} be a path from f_{k}(x) to x for k = 0, 1. Let v be the path h(x, −) from f_{0}(x) to f_{1}(x). Write π = π_{1}(X, x). Let ϕ_{k} for k = 0, 1 be the endomorphism of π and ℤπ respectively given by c_{w_{k}} ∘ π_{1}(f_{k}, x). Notice that ϕ_{1} = c_{w_{o}^{−}∗v∗w_{1}} ∘ ϕ_{0}. Hence we obtain a map

Θ(x, v) : Θ(ℤπ, Φ_{0}) ⟶ Θ(ℤπ, ϕ_{1}),  (2.9)

which sends the class of g : F ⟶ ϕ_{0}^{∗}F to the class of l_{w_{o}^{−}∗v∗w_{1}} ∘ g : F ⟶ ϕ_{1}^{∗}F, where l_{w_{o}^{−}∗v∗w_{1}} : F ⟶ c_{v}^{∗}F is the ℤπ-map sending x to (w_{o}^{−} ∗ v ∗ w_{1})x. One easily checks using homotopy invariance

Θ(x, v)(u(f_{0}, x_{0}, w_{0})) = u(f_{1}, x_{1}, w_{1})  (2.10)

and that Θ_{h} defined in 2.8 and Θ(x, v) defined in 2.9 agree under the obvious identifications.

Now suppose further that f_{0} and f_{1} agree, and for simplicity f_{0}(x) = x. Write f = f_{0} = f_{1}. Define Jiang’s subgroup

J(f, x) ⊂ π  (2.11)

as the subgroup of elements for which there is a homotopy h from f to f such that v is represented by h(x, −). Choose w_{0} = w_{1} to be the trivial path. Then ϕ_{0} = ϕ_{1} is just ϕ := π_{1}(f, x). For v ∈ J(f, x) the map Θ(x, v) defined in 2.9 becomes the automorphism

U(ℤπ, ϕ) ⟶ U(ℤπ, ϕ) [g : F ⟶ ϕ^{∗}F] ↦ [l_{v} ∘ g : F ⟶ ϕ^{∗}F].

Hence J(f, x) acts on U(ℤπ) and

u(f) ∈ U(ℤπ, ϕ)^{J(f,x)}.  (2.12)

Jiang’s subgroup is studied for instance in [2], [17] and [18] and all the results there about Nielsen numbers can be derived from 2.12 and Theorem 2.5 since they imply in the notation of Example 3.5

λ(f) ∈ Λ(ℤπ, ϕ)^{J(f,x)},

where u ∈ J(f, x) acts on [x] ∈ Λ(ℤπ, ϕ) by u[x] = [ux].

3 Examples

In this section we explain that the universal invariant defined in 2.2 encompasses some of the known Lefschetz type invariants and others.

Example 3.1. The (classical) Lefschetz number of an endomorphism f : X ⟶ X of a finite CW -complex is defined as the integer

λ_{class}(f) := ∑_{n≥0}(−1)^{n} · tr(H_{n}(f; ℤ)) ∈ ℤ,

where tr(H_{n}(f; ℤ)) ∈ ℤ is the trace of the endomorphism H_{n}(f; ℤ) of the finitely generated abelian group H_{n}(X; ℤ). Recall that the trace of an endomorphism of a finitely generated abelian group A is the trace of the integer square matrix given by the induced endomorphism of the finitely generated free abelian group A/ Tors(A) with respect to some basis of A/ Tors(A). If one takes Θ to be the constant functor with value ℤ and θ(f) = λ_{class}(f), one obtains a functorial Lefschetz invariant for finite CW -complexes.

The unique natural transformation

ξ(f) : U(f) ⟶ ℤ

which sends u(f) to λ_{class}(f) (see Definition 2.4 and Theorem 2.5) is given by the homomorphism

U(ℤπ_{1}(X, x), ϕ(f, w, x)) ⟶ ℤ  (3.2)

which maps the class of the endomorphism g : F ⟶ ϕ^{∗}F to the trace of the endomorphism of finitely generated free abelian groups

\overline{g} : ℤ ⊗_{ℤπ_{1}(X,x)} F ⟶ ℤ ⊗_{ℤπ_{1}(X,x)} F n ⊗ x ↦ n ⊗ f(x).

Example 3.3. Let f : X ⟶ X be an endomorphism of a finite CW -complex X. Let det(I − t · H_{n}(f; ℚ)) be the characteristic polynomial of the endomorphism H_{n}(f; ℚ) of the rational vector space H_{n}(X; ℚ). Define the rational Lefschetz function

r(f) := ∏_{n≥0} det(I − t · H_{n}(f; ℚ))^{(−1)^{n}}

which is a rational function in t with rational coefficients. If we take Θ to be the constant functor with the abelian group rat(t) of rational functions in t and θ(f) to be r(f) we obtain a functorial Lefschetz invariant.

Define the Lefschetz Zeta-function to be the formal power series in t with rational coefficients

ζ(f) := ∑_{k≥1}^{∞} t^{k} \frac{Λ_{class}(f^{k})}{k}.

If we take Θ to be the constant functor with the abelian group of formal power series in t with rational coefficients and θ(f) to be ζ(f), we obtain a functorial Lefschetz invariant. These two invariants are related by the following formulas of formal power series (see [38, section 3])

r(f) = exp(−ζ(f));
ζ(f) = − ln(r(f)).

The natural transformation

ξ(f) : U(f) ⟶ rat(t)

which sends u(f) to r(f) (see Definition 2.4 and Theorem 2.5) is given by the homomorphism

U(ℤπ_{1}(X, x), ϕ(f, w, x)) ⟶ rat(t)  (3.4)

which maps the class of the endomorphism g : F ⟶ ϕ^{∗}F to the characteristic polynomial det(I − t\overline{g}) of the endomorphism of finitely generated free abelian groups

\overline{g} : ℤ ⊗_{ℤπ_{1}(X,x)} F ⟶ ℤ ⊗_{ℤπ_{1}(X,x)} F n ⊗ x ↦ n ⊗ f(x).

The classical Lefschetz function, variations of it and their relation to Reidemeister torsion have been investigated for instance in [11], [12], [13], [14].

Example 3.5. Next we recall the definition of the (generalized) Lefschetz invariant (see [41], [42]). Let f : X ⟶ X be an endomorphism of a finite CW -complex. Assume for a moment that X is connected. Fix a base point x ∈ X and a path w from y = f(x) to x. Now make the following additional choices of a model of the universal covering p : X̃ ⟶ X and of a base point x̃ ∈ X̃ with p(x̃) = x. Let ỹ ∈ X̃ be the point uniquely characterized by the property that w lifts to a path in X̃ from ỹ to x̃. There is precisely one lifting f̃: X̃ ⟶ X̃ with f̃(x̃) = ỹ. Abbreviate π = π_{1}(X, x). We obtain a homomorphism of groups ϕ : π ⟶ π by the composition of the map π_{1}(f, x) : π_{1}(X, x) ⟶ π_{1}(X, y) induced by f and the map π_{1}(X, y) ⟶ π_{1}(X, x) given by conjugation with w. In the sequel we let operate π on X̃ from the left where the operation is defined with respect to the base point x̃ ∈ X̃ (see Section 2). Then the map f̃ is ϕ-equivariant. It induces a ℤπ-chain map C(f̃) : C(X̃) ⟶ ϕ^{∗}C(X̃) on the cellular ℤπ-chain complex of X̃.

Let g : F ⟶ ϕ^{∗}F be a ℤπ-map. Choose a ℤπ-basis {b_{1}, … b_{k}} for F. Let A = (A_{i,j}) be the square (r, r)-matrix over ℤπ of g with respect to the chosen basis, i.e. g(b_{i}) = ∑_{j} A_{i,j}b_{j}. Let ℤπ_{ϕ} be the abelian group which is the quotient of the abelian group ℤπ by the abelian subgroup generated by all elements of the form ϕ(v)w − wv for all v, w ∈ π. We call two elements w_{0}, w_{1} ∈ π ϕ-conjugated if there is u ∈ π with ϕ(u)w_{0}u^{−1} = w_{1}. This is an equivalence relation on π and ℤπ_{ϕ} can be identified with free abelian group generated by the ϕ-conjugacy classes [w] of elements in w in π. For an element x ∈ ℤπ let [x] ∈ ℤπ_{ϕ} be its image under the obvious projection ℤπ ⟶ ℤπ_{ϕ}. Define

tr_{(ℤπ,ϕ)}(g) := ∑_{i}[A_{i,i}]  ∈ ℤπ_{ϕ}  (3.6)

and

Λ(f, x, w) := ℤπ_{ϕ};  (3.7)
λ(f, x, w) := ∑_{n≥0}(−1)^{n} · tr_{(ℤπ,ϕ)}(C_{n}(f̃))  ∈ Λ(f, x, w).  (3.8)

One easily checks the invariant and the group it takes values in are independent of the choice of p, of x̃ ∈ p^{−1}(x) and the bases but it depends on the choice of x and the homotopy class relative end points of the path w. Let x′ and w′ be a second choice. Let v be any path from x to x′. We obtain by conjugation with w a map π_{1}(X, x) ⟶ π_{1}(X, x′) which induces a map μ(x, w, x′, w′) : Λ(f, x, w) ⟶ Λ(f, x′, w′). This map is indeed independent of the choice of v, sends λ(f, x, w) to λ(f, x′, w′) and satisfies μ(x′, w′, x′′, w′′) ∘ μ(x, w, x′, w′) = μ(x, w, x′′, w′′) and μ(x, w, x, w) = id. Now define Λ(f) as the abelian group which is the set of equivalence classes of the equivalence relation on ∐_{(x,w)} Λ(f, x, w) generated by u ∼ μ(x, w, x′, x, w′)(u) for u ∈ Λ(f, x, w). The collection of the λ(f, x, w) determines an element

λ(f) ∈ Λ(f).  (3.9)

The obvious map Λ(f, w, x) ⟶ Λ(f) is an isomorphism and sends λ(f, x, w) to λ(f) for all (x, w).

If X has more than one path component, one defines Λ(f) by the direct sum over the path components C of X of the groups Λ(f|_{C}). Recall that we require that f induces the identity on π_{0}(X). Define λ(f) ∈ Λ(f) analogously. Then (Λ, λ) is a functorial Lefschetz invariant for finite CW -complexes. Notice that in contrast to the previous Examples 3.1 and 3.3 Λ is not a constant functor.

Consider as example the endomorphism f_{d} : S^{1} ⟶ S^{1} sending z to z^{d} for d ∈ ℤ. Let ℤ/(|d − 1|) be the cyclic group of order |d − 1| if d ≠ 1 and of infinite order if d = 1. Let t be the image of the generator of ℤ, written multiplicatively, under the canonical projection onto ℤ/(|d − 1|). Then there is an obvious isomorphism

Λ(S^{1}, f_{d}) ⟶^{≅} ℤ[ℤ/(|d − 1|)]

which sends λ(f_{d}) to − ∑_{k=1}^{d−1} t^{k} if d ≥ 2, to ∑_{k=0}^{|d|} t^{−k} if d ≤ 0 and to 0 if d = 1.

We call λ(f) ∈ Λ(f) the (generalized) Lefschetz invariant. The Nielsen number of f is the number of ϕ-conjugacy classes of elements in ℤπ_{1}(X, x)_{ϕ} which appear with non-trivial coefficients in λ(f). The Nielsen number and the generalized Lefschetz invariant of f vanish if f is homotopic to an endomorphism without fixed points. Any endomorphism homotopic to f has at least N(f)-fixed points. Moreover, f is homotopic to an endomorphism with precisely N(f) fixed points and f is homotopic to a map without fixed points if and only if λ(f) and N(f) vanish, provided that X is a compact manifold possibly with boundary of dimension different from 2. Next we recall the Lefschetz fixed point formula. Suppose that f : X ⟶ X is an endomorphism of a connected compact manifold possibly with boundary such that f has only finitely many fixed points z which do not lie on ∂X and satisfy det(id −T_{z}f) ≠ 0 where T_{z}f : T_{z}X ⟶ T_{z}X is the differential of f at z. Then

λ(f, x, w) = ∑_{z∈F ix(f)} \frac{det(id −T_{z}f)}{| det(id −T_{z}f)|} · [u_{z} ∗ f(u_{z})^{−1} ∗ w],  (3.10)

where u_{z} is any path from x to z. For further information we refer for instance to [2], [7], [11], [12], [18], [19], [20] [21].

The unique natural transformation

ξ(f) : U(f) ⟶ Λ(f)

which sends u(f) to λ(f) (see Definition 2.4 and Theorem 2.5) is given by the homomorphism

U(ℤπ_{1}(X, x), ϕ(f, w, x)) ⟶ ℤπ_{1}(X, x)_{ϕ(f,w,x))}  [g] ↦ tr_{(ℤπ,ϕ)}(g).  (3.11)

The next example does not seem to be covered by classical Lefschetz-type invariants.

Example 3.12. Let f : X ⟶ X be an endomorphism of a finite connected CW -complex. The mapping torus T_{f} is obtained from the cylinder X ×I by identifying the bottom and top using f. If f and g are homotopic then their mapping tori are simple homotopy equivalent (see [6]). Hence a simple homotopy invariant of T_{f} is an invariant of the homotopy class of f. For instance one can interpret r(f) introduced in Example 3.3 as the Reidemeister torsion of the canonical infinite cyclic covering of T_{f} [38, section 3]. One can also apply a more sophisticated invariant to T_{f}, namely the combinatorial L^{2}-torsion. It is known that the L^{2}-Betti numbers of T_{f} all vanish [30, Theorem 2.1]. We assume in the sequel that T_{f} is of determinant class in the sense of [4, page 754], we discuss this assumption later. Then the combinatorial L^{2}-torsion is defined (see for instance [5], [29], [31])

ρ^{(2)}(T_{f}) ∈ ℝ.

If X is a compact 2-dimensional manifold and f a diffeomorphism, then the L^{2}-torsion of T_{f} can be computed in terms of the volumes of the hyperbolic pieces in its decomposition by a minimal family of pairwise non-isotopic incompressible not boundary-parallel embedded 2-tori into Seifert pieces and hyperbolic pieces. This is proven in [35] using [3], [23] and [37]. In particular if f : F ⟶ F is a pseudo-Anosov selfhomeomorphism of a closed hyperbolic 2-dimensional manifold, then the mapping torus T_{f} is a closed hyperbolic 3-manifold and its combinatorial L^{2}-torsion is −1/3π times its volume.

There is a natural homomorphism

ρ(f) : U(f) ⟶ ℝ  (3.13)

which sends an endomorphism g : F ⟶ ϕ^{∗}F of finitely generated free ℤπ_{1}(X)-module F to the generalized Fuglede-Kadison-determinant of the endomorphism of finitely generated Hilbert N (π_{1}(T_{f}))-modules

\overline{g} : l^{2}(π_{1}(T_{f})) ⊗_{ℤπ_{1}(X)} F ⟶ l^{2}(π_{1}(T_{f})) ⊗_{ℤπ_{1}(X)} F u ⊗ v ↦ −ut ⊗ g(v) + u ⊗ v

in the sense of [29, Section 4]. Here t ∈ π_{1}(T_{f}) is the element given by the composition of the path [0, 1] ⟶ T_{f} s ↦ (s, x) with some path in X from f(x) to x for some basepoint x ∈ X. Because of the computation of the cellular ℤπ_{1}(T_{f})- chain complex of T̃_{f} in [30, page 207], ρ has the property

ρ(f)(u(f)) = ρ^{(2)}(T_{f}).

We see that u(f) determines ρ^{(2)}(T_{f}). However, the pair (ℝ, ρ^{(2)}(T_{?})), which consists of the constant functor with value ℝ and the function sending f to ρ^{(2)}(T_{f}) is not quite a functorial Lefschetz invariant because Additivity holds only for those push outs for which for k = 0, 1, 2 and any base point x_{k} ∈ X_{k} the map π_{1}(j_{k}, x_{k}) : π_{1}(X_{k}, x_{k}) ⟶ π_{1}(X, j_{k}(x_{k})) is injective [29, Theorem 1.6]. All other axioms are satisfied in full generality.

Next we discuss the assumption of determinant class which is needed to define L^{2}- torsion or generalized Fuglede-Kadison determinant. Notice that a finite CW -complex X is of determinant class if all its Novikov-Shubin-invariants are positive and that there is the conjecture that the Novikov-Shubin invariant of any finite CW -complex are positive [24, Conjecture 7.2], but it is known only in special cases like T_{f} in case of an endomorphism of a compact surface. If π_{1}(X) is residually finite or amenable respectively, then π_{1}(T_{f}) is residually finite or amenable respectively and T_{f} is of determinant class and 3.13 is welldefined. (see [3, Theorem A in Appendix A] and [8, Theorem 0.2]).

Remark 3.14. As always in algebraic K-theory it is often useful for computations for group rings to use representations to detect elements. This strategy applies also in our context.

Let f : X ⟶ X be an endomorphism of a connected finite CW -complex X. Let A be a commutative ring and V be a right Aπ-module such that V as a A-module is finitely generated free. Let t : V ⟶ ϕ^{∗}V be a Aπ-map. Then we obtain a homomorphism

R_{V,t} : U(f) ⟶ A  (3.15)

by sending the class of the ℤπ-map g : F ⟶ ϕ^{∗}F to the trace of the endomorphism of finitely generated free A-modules given by the composition

V ⊗_{ℤπ} F ⟶ V ⊗_{ℤπ} F v ⊗ f ↦ t(v) ⊗ g(f).

Computations using representations are given in [21]. We explain its relation to the mapping torus approach in Remark 7.14.

Other constructions of Lefschetz type invariants taking values in Hochschild-homology and A-theory are given in [15] and [36].

There are higher analogues of the groups U(R, ϕ), just apply the standard constructions of Quillen or Waldhausen to the category of ϕ-endomorphisms of finitely generated free R-modules. Analogously one can define an A-theoretic version of the geometric side of endomorphisms of finite CW -complexes and construct a linearization map from the Atheory version to the K-theory version analogously to the linearization map from A(X) to K(ℤπ_{1}(X)) for a connected finite CW -complex X.

4 Proof of the universal property

This section is devoted to the proof of Theorem 2.5. For this purpose we will need the following notions and constructions.

Let X be a space. A retractive space over X is a triple Y = (Y, i, r) which consists of a space Y, a cofibration i : X ⟶ Y and a map r : Y ⟶ X satisfying r ∘ i = id. We often identify X with i(X). Given a retractive space Y over X, define retractive spaces Y ×_{X} [0, 1] and C_{X}Y by the push outs

X × [0, 1] ⟶^{pr}  X
i×id↓  ↓
Y × [0, 1] ⟶ Y ×_{X} [0, 1]

and

Y × {1} ⟶^{r} X
j↓  ↓
Y ×_{X} [0, 1] ⟶ C_{X}Y

where pr resp. j is the canonical projection resp. inclusion and the inclusion of X and the retraction onto X is the obvious one. Define the retractive space Σ_{X}Y by the push out

Y \underrightarrow{î} C_{X}Y
î↓  ↓
C_{X}Y ⟶ Σ_{X}Y

where î: Y ⟶ C_{X}Y is the inclusion induced by the inclusion Y × {0} ⟶ Y × [0, 1]. Notice that the composition î∘ i : X ⟶ C_{X}Y is a homotopy equivalence relative X with the retraction of C_{X}Y onto X as homotopy inverse relative X. If X consists of one point, then a retractive space over X is just a pointed space and C_{X}Y resp. Σ_{X}Y is the reduced cone resp. suspension of Y.

Given two retractive spaces Y and Z over X and an endomorphism f : X ⟶ X, define [(C_{X}Y, Y), (C_{X}Z, Z)]_{f} to be the set of homotopy classes relative X of maps of pairs (ĝ, g) : (C_{X}Y, Y) ⟶ (C_{X}Z, Z) which induce on X the given endomorphism f. Homotopy class relative X means that the relevant homotopies are stationary on X. Next we want to describe a suspension map

Σ_{X} : [(C_{X}Y, Y), (C_{X}Z, Z)]_{f} ⟶ [(C_{X}Σ_{X}Y, Σ_{X}Y), (C_{X}Σ_{X}Z, Σ_{X}Z)]_{f}.  (4.1)

Let (ĝ, g) be a representative of a class in the source. We only explain the definition of a representative (ĝ_{Σ}, g_{Σ}) of the image of the class under this map. Define g_{Σ} by ĝ ∪_{g} ĝ. Notice that C_{X} is compatible with push outs so that we can think of C_{X}Σ_{X}Y as the push out of C_{X} applied to the diagram defining Σ_{X}, i.e.

C_{X}Y ⟶^{C_{X}î} C_{X}C_{X}Y
C_{X}î↓  ↓
C_{X}C_{X}Y ⟶ C_{X}Σ_{X}Y

In order to define the extension ĝ_{Σ} : C_{X}Σ_{X}Y ⟶ C_{X}Σ_{X}Y we will define an endomorphism \overline{g} : C_{X}C_{X}Y ⟶ C_{X}C_{X}Y extending ĝ and will put ĝ_{Σ} to be \overline{g} ∪ĝ \overline{g}. For the definition of \overline{g} it is convenient to rewrite C_{X}C_{X}Y as follows. Namely, there is a commutative diagram

Y × [0, 1] × [0, 1] ⟶^{ψ} Y × [0, 1] × [0, 1]
↓  ↓
C_{X}Y × [0, 1] ∪_{Y ×[0,1]} Y ⟶^{\overline{ψ}}  C_{X}C_{X}Y

where

ψ(y, s, t) = (y, \frac{ts}{max\{t, 1 − t\}}, \frac{(1 − t)s}{max\{t, 1 − t\}}) ,

the vertical arrows are the obvious projections and the space in the left lower corner is the push out of C_{X}Y × [0, 1] ⟵^{j} Y × [0, 1] ⟶^{pr} Y for j resp. pr the canonical inclusion resp. projection. One easily checks that \overline{ψ} is a homeomorphism. Conjugating the endomorphism ĝ × id ∪_{g×id}g with \overline{ψ} yields \overline{g}. This finishes the definition 4.1 of the map Σ_{X}.

Let f : X ⟶ X be an endomorphism of a finite CW -complex. Suppose for some time that X is connected. Let p : X̃ ⟶ X be a model of the universal covering, x ∈ X and x̃ ∈ X̃ base points with p(x̃) = x, w a path from f(x) to x and f̃: X̃ ⟶ X̃ the lift of f for which w lifts to a path from f̃(x̃) to x̃. Given a retractive space Y, let Ỹ and { \atop C_{X}Y} be the pull back of X̃ with the retractions onto X. We call a retractive space Y over X a d-extension if Y is obtained from X by attaching finitely many cells in dimension d. If Y is a d-extension of X and d ≥ 2, then π_{1}(r) is an isomorphism and Ỹ is the universal covering of Y. Given a map g : Y ⟶ Z of d-extensions over X for d ≥ 2 which induces f on X, define g̃: Ỹ ⟶ Z̃ to be the lift of g uniquely determined by the property that it induces f̃ on X̃. Let ϕ : ℤπ_{1}(X, x) ⟶ ℤπ_{1}(X, x) be the homomorphism induced as before by the composition c_{w} ∘ π_{1}(f, x).

We have already seen that U is the universal additive invariant for chain complexes. The reason why it turns out to be the universal invariant for spaces is the next lemma which contains the decisive step in the passage from geometry to algebra.

Lemma 4.2. Let Y and Z be d-extensions of X for d ≥ 2. Then the map

η : [(C_{X}Y, Y), (C_{X}Z, Z)]_{f} ⟶ hom_{ℤπ_{1}(X,x)}(C_{d}(Ỹ, X̃), ϕ^{∗}C_{d}(Z̃, X̃)) [(ĝ, g)] ↦ C_{d}(g̃, f̃)

is bijective.

Proof. Choose a push out

∐_{i∈I} S^{d−1} \underrightarrow{∐_{i∈I} q_{i}} X
k↓  ↓^{j}
∐_{i∈I} D^{d} \underrightarrow{∐_{i∈I} Q_{i}} Y

Define a map p_{i} : S^{d} ⟶ Y by requiring that p_{i} on the upper hemisphere S_{+}^{d} is Q_{i} and on the lower hemisphere S_{−}^{d} is r ∘ Q_{i}. Then C_{X}Y is the push out

∐_{i∈I} S^{d} \underrightarrow{∐_{i∈I} p_{i}} Y
k↓  ↓ĵ
∐_{i∈I} D^{d+1} \underrightarrow{∐_{i∈I} P_{i}} C_{X}Y

Hence a map (ĝ, g) : (C_{X}Y, Y) ⟶ (C_{X}Z, Z) with g|_{X} = f is uniquely determined by its compositions (ĝ, g) ∘ (P_{i}, p_{i}) and any collection of maps (k̂_{i}, k_{i}) : (D^{d+1}, S^{d}) ⟶ (C_{X}Z, Z) with k_{i}|_{S_{−}^{d}} = f ∘ r ∘ Q_{i} determines uniquely such a map (ĝ, g) with (ĝ, g) ∘ (P_{i}, p_{i}) = (k̂_{i}, k_{i}). Hence the map

μ : [(C_{X}Y, Y), (C_{X}Z, Z)]_{f} ⟶ ∏_{i∈I}[(D^{d+1}, S^{d}), (C_{X}Z, Z)]_{f∘r∘Q_{i}:S_{−}^{d}⟶X}.

sending [ĝ, g] to ([ĝ, g) ∘ (P_{i}, p_{i})])_{i∈I} is a bijection.

Next we show that the following forgetful map is bijective for i ∈ I:

ν_{i} : [(D^{d+1}, S^{d}), (C_{X}Z, Z)]_{f∘r∘Q_{i}:S_{−}^{d}⟶X} ⟶ π_{d+1}(C_{X}Z, Z, f(x_{i})),

where s = (0, 0, …, −1) ∈ S^{d} is a fixed base point and x_{i} = r∘Q_{i}(s). It forgets that the map has to look like f ∘ r ∘ Q_{i} on the lower hemisphere and remembers only that the point s ∈ S^{d} has to go to f(x_{i}). It is bijective as the inclusion {s} ⟶ S_{−}^{d} is a homotopy equivalence and the inclusions {s} ⟶ S_{−}^{d} ⟶ S^{d} ⟶ D^{d+1} are cofibrations. Choose paths w_{i} from x_{i} to x in Y. Composing each ν_{i} with the isomorphism π_{d+1}(C_{X}Z, Z, f(x_{i})) ⟶ π_{d+1}(C_{X}Z, Z, f(x)) given by the path f(w_{i}) from f(x_{i}) to f(x) and taking the product of the resulting isomorphisms yields an isomorphism

ν : ∏_{i∈I}[(D^{d+1}, S^{d}), (C_{X}Y, Y)]_{f∘r∘Q_{i}:S_{−}^{d}⟶X} ⟶ ∏_{i∈I} π_{d+1}(C_{X}Z, Z, f(x)).

Hence the composition

ν ∘ μ : [(C_{X}Y, Y), (C_{X}Z, Z)]_{f} ⟶ ∏_{i∈I} π_{d+1}(C_{X}Z, Z, f(x))

is bijective.

Each pair (Q_{i}, q_{i}) : (D^{d+1}, S^{d}) ⟶ (C_{X}Y, Y) defines an element in π_{d+1}(C_{X}Y, Y, x_{i}). Denote its image under the map induced by the path w_{i} by b_{i} ∈ π_{d+1}(C_{X}Y, Y, x). Then {b_{i} | i ∈ I} is ℤπ_{1}(X, x)-basis for π_{d+1}(C_{X}Y, Y, x). One easily checks that the bijective composition ν ∘ μ above is given by

[(C_{X}Y, Y), (C_{X}Z, Z)]_{f} ⟶ ∏_{i∈I} π_{d+1}(C_{X}Z, Z, f(x)) [(ĝ, g] ↦ (π_{d+1}(ĝ, g, x)(b_{i}))_{i∈I}.

Consider the following isomorphism given by the following composition of isomorphisms or their inverses

σ_{Y} : π_{d+1}(C_{X}Y, Y, x) ⟶^{∂} π_{d}(Y, X, x) ⟵^{π_{d}(p,x̃)} π_{d}(Ỹ, X̃, x̃) ⟶^{h} H_{d}(Ỹ, X̃) = C_{d}(Ỹ, X̃).

Here ∂ is the boundary operator in the long exact homotopy sequence of the triple (C_{X}Y, Y, X) which is an isomorphism since the inclusion X ⟶ C_{X}Y is a homotopy equivalence and h is the Hurewicz isomorphism. Analogously define the isomorphism

σ_{Z} : π_{d+1}(C_{X}Z, X, f(x)) ⟶ C_{d}(Z̃, X̃).

For a map (ĝ, g) : (C_{X}Y, Y) ⟶ (C_{X}Z, Z,) with g|_{X} = f the following diagram commutes

π_{d+1}(C_{X}Y, Y, x) ⟶^{c_{w}∘π_{d+1}(ĝ,g,x)} π_{d+1}(C_{X}Z, Z, f(x))
σ_{Y} ↓  ↓^{σ_{Z}}
C_{d}(Ỹ, X̃)  ⟶^{C(g̃,f̃)}  C_{d}(Z̃, X̃)

Now Lemma 4.2 follows.

Given a functorial Lefschetz invariant (Θ, θ), we want to construct for an endomorphism f : X ⟶ X of a connected finite CW -complex and a d-extension Y a map

τ_{Y} : [(C_{X}Y, Y), (C_{X}Y, Y)]_{f} ⟶ Θ(X, f) [ĝ, g)] ↦ Θ(î∘ i)^{−1} ∘ Θ(î)(θ(Y, g)). (4.3)

Recall that i : X ⟶ Y and î: Y ⟶ C_{X}Y are the inclusions and î∘ i is a homotopy equivalence so that Θ(î∘ i) is bijective because of invariance under homotopy equivalence. We have to show that Θ(î∘ i)^{−1} ∘ Θ(î)(θ(Y, g)) depends only on the homotopy class of (ĝ, g) relative X. Let (Ĝ, G) be such a homotopy relative X between (ĝ_{0}, g_{1}) and (ĝ_{1}, g_{1}). For k = 0, 1 let l_{k} be the inclusion X ⟶ X × [0, 1] sending y to (y, k) (and similiar for Y and C_{X}Y). The following diagram in End(C) commutes:

Θ(X, f) ⟶^{Θ(l_{0})} Θ(X × [0, 1], f × id) ⟵^{Θ(l_{1})} Θ(X, f)
Θ(i)↓  ^{Θ(i×id)}↓  ^{Θ(i)}↓
Θ(Y, g_{0}) ⟶^{Θ(l_{0})} Θ(Y × [0, 1], G × pr_{[0,1]}) ⟵^{Θ(l_{1})} Θ(Y, g_{1})
Θ(î)↓  ^{Θ(î×id)}↓  ^{Θ(î)}↓
Θ(C_{X}Y, ĝ_{0}) ⟶^{Θ(l_{0})} Θ(C_{X}Y × [0, 1], Ĝ × pr_{[0,1]}) ⟵^{Θ(l_{1})} Θ(C_{X}Y, ĝ_{1})

Notice that Θ(l_{0}) = Θ(l_{1}) : Θ(X, f) ⟶ Θ(X × [0, 1], f × id) holds because of the homotopy invariance of Θ. We have Θ(l_{k})(θ(g_{k})) = θ(G × pr_{[0,1]}) for k = 0, 1 because of invariance under homotopy equivalence. Now a simple diagram chase shows that Θ(î∘ i)^{−1} ∘ Θ(î)(θ(Y, g_{k})) is independent of k = 0, 1. Hence the map τ_{Y} of 4.3 is well-defined.

Next we want to define a map

τ_{d} : Ũ(X, f) ⟶ Θ(X, f)  (4.4)

for an integer d ≥ 2. We will construct a map

τ_{d} : Ũ(ℤπ_{1}(X, x), ϕ) ⟶ Θ(X, f)

for a fixed choice of base point x ∈ X and path w from f(x) to x and leave it to the reader that the maps for the various choices fit together to give the desired map. Let F be a finitely generated free ℤπ-module and a : F ⟶ ϕ^{∗}F a ℤπ_{1}(X, x)-endomorphism. Choose a d-extension Y together with an ℤπ_{1}(X, x)-isomorphism b : F ⟶ C_{d}(Ỹ, X̃). Such Y exists, namely, take the wedge of X with a finite number of copies of S^{d} with the obvious retraction onto X. Then b ∘ a ∘ b^{−1} is an element in hom_{ℤπ_{1}(X,x)}(C_{d}(Ỹ, X̃), ϕ^{∗}C_{d}(Ỹ, X̃)). Let [(ĝ, g)] be its preimage under the isomorphism

η : [(C_{X}Y, Y), (C_{X}Y, Y)]_{f} ⟶ hom_{ℤπ_{1}(X,x)}(C_{d}(Ỹ, X̃), ϕ^{∗}C_{d}(Ỹ, X̃))

of Lemma 4.2. Define

τ_{d}([a]) = (−1)^{d} · (τ_{Y} ([ĝ, g]) − θ(f)) .

We must show that this is independent of the choice of Y and b. Suppose Z and c is a second choice yielding an element [(ĥ, h)] in [(C_{X}Z, Z), (C_{X}Z, Z)]_{f}. Let (k̂, k) be the preimage of c ∘ b^{−1} under the isomorphism

η : [(C_{X}Y, Y), (C_{X}Z, Z)]_{id} ⟶ hom_{ℤπ_{1}(X,x)}(C_{d}(Ỹ, X̃), C_{d}(Z̃, X̃))

of Lemma 4.2 and (k̂^{−1}, k^{−1}) be the preimage of b ∘ c^{−1} under the isomorphism

η : [(C_{X}Z, Z), (C_{X}Y, Y)]_{id} ⟶ hom_{ℤπ_{1}(X,x)}(C_{d}(Z̃, X̃), C_{d}(Ỹ, X̃))

of Lemma 4.2. We conclude from invariance under homotopy equivalence applied to the morphism k : (Y, k^{−1} ∘ h ∘ k) ⟶ (Z, k ∘ k^{−1} ∘ h)

τ_{Y} ((k̂^{−1}, k^{−1}) ∘ (ĥ, h) ∘ (k̂, k)) = τ_{Z}((k̂, k)) ∘ (k̂^{−1}, k^{−1}) ∘ (ĥ, h)).

Since (k̂^{−1}, k^{−1}) ∘ (ĥ, h) ∘ (k̂, k)) and (ĝ, g) define the same class in [(C_{X}Y, Y), (C_{X}Y, Y)]_{f} because of Lemma 4.2, we get:

τ_{Y} ((k̂^{−1}, k^{−1}) ∘ (ĥ, h) ∘ (k̂, k)) = τ_{Y} (ĝ, g).

Similarly we get

τ_{Z}((k̂, k)) ∘ (k̂^{−1}, k^{−1}) ∘ (ĥ, h)) = τ_{Z}(ĥ, h).

This shows that the definition of τ_{d}(a) is independent of the choice of the d-extension Y and the isomorphism b. Hence we have defined τ_{d} on generators. It remains to check that τ_{d} is compatible with the relations appearing in the definition of U(ℤπ_{1}(X, x), ϕ).

We have to show for a block endomorphism

a = ( \begin{matrix}a_{0} & a_{2} \\ 0 & a_{1} \end{matrix} ) : F_{0} ⊕ F_{1} ⟶ ϕ^{∗}F_{0} ⊕ ϕ^{∗}F_{1}

that τ_{d}(a) = τ_{d}(a_{0}) + τ_{d}(a_{1}) holds. Choose d-extensions Y_{k} and ℤπ_{1}(X, x)-isomorphisms b_{k} : F_{k} ⟶ C_{d}(Ỹ_{k}, X̃) for k = 0, 1. Put Y = Y_{0} ∪_{X} Y_{1}. Notice that C_{X}Y = C_{X}Y_{0} ∪_{X} C_{X}Y_{1}, i = i_{0} ∪_{X} i_{1} and î = î_{0} ∪_{X} î_{1} holds where i_{k} : X ⟶ Y_{k}, i : X ⟶ Y, î_{k} : Y_{k} ⟶ C_{X}Y_{k} and î: Y ⟶ C_{X}Y are the canonical inclusions. Then the direct sum of b_{0} and b_{1} yields a ℤπ_{1}(X, x)-isomorphism b : F_{0} ⊕ F_{1} ⟶ C_{d}(Ỹ, X̃). Let (ĝ, g) : (C_{X}Y, Y) ⟶ (C_{X}Y, Y) be a map such that it induces a map (ĝ_{0}, g_{0}) : (C_{X}Y_{0}, Y_{0}) ⟶ (C_{X}Y_{0}, Y_{0}) and under the identifications b_{0} and b above the induced chain endomorphisms agree with a_{0} and a. Denote by j : (C_{X}Y_{1}, Y_{1}) ⟶ (C_{X}Y_{0} ∪_{X} C_{X}Y_{1}, C_{X}Y_{0} ∪_{X} Y_{1}) the canonical inclusion and denote by p : (C_{X}Y_{0} ∪_{X} C_{X}Y_{1}, C_{X}Y_{0} ∪_{X} Y_{1}) ⟶ (C_{X}Y_{1}, Y_{1}) the canonical projection. Because of the identification C_{X}Y_{0} ∪_{X} Y_{1} = C_{X}Y_{0} ∪_{Y_{0}} (Y_{0} ∪_{X} Y_{1}) we can view (ĝ, ĝ_{0} ∪_{g_{0}} g) as an endomorphism of (C_{X}Y_{0} ∪_{X} C_{X}Y_{1}, C_{X}Y_{0} ∪_{X} Y_{1}). Then the endomorphism p ∘ (ĝ, ĝ_{0} ∪_{g_{0}} g) ∘ j induces under the identification b_{1} the map a_{1}. We conclude from the definition of τ_{d}

τ_{d}(a_{0}) = Θ(î_{0} ∘ i_{0})^{−1} ∘ Θ(î_{0})(θ(g_{0})) − θ(f)
τ_{d}(a) = Θ(î∘ i)^{−1} ∘ Θ(î)(θ(g)) − θ(f)
τ_{d}(a_{1}) = Θ(î_{1} ∘ i_{1})^{−1} ∘ Θ(î_{1})(θ(p ∘ (ĝ_{0} ∪_{g_{0}} g) ∘ j)) − θ(f).

Applying additivity to the push out diagram

(Y_{0}, g_{0}) ⟶^{id_{Y_{0}} ∪_{X}i_{1}}  (Y_{0} ∪_{X} Y_{1}, g)
î_{0}↓  ↓^{î_{0}∪_{X}id_{Y_{1}}}
(C_{X}Y_{0}, ĝ_{0}) ⟶^{id_{C_{X} Y_{0}} ∪_{X}i_{1}} (C_{X}Y_{0} ∪_{Y_{0}} (Y_{0} ∪_{X} Y_{1}), ĝ_{0} ∪_{g_{0}} g)

and the identification C_{X}Y_{0} ∪_{X} Y_{1} = C_{X}Y_{0} ∪_{Y_{0}} (Y_{0} ∪_{X} Y_{1}) yields

θ(ĝ_{0} ∪_{g_{0}} g) = Θ(î_{0} ∪_{X} id_{Y_{1}})(θ(g)) + Θ(id_{C_{X}Y_{0}} ∪_{X}i_{1})(θ(ĝ_{0})) − Θ((id_{C_{X}Y_{0}} ∪_{X}i_{1} ∘ î_{0})(θ(g_{0})).

Applying Θ(î∘ i)^{−1} ∘ Θ(id_{C_{X}Y_{0}} ∪_{X} î_{1}) to this formula and invariance under homotopy equivalence applied to î_{0} : (X, f) ⟶ (C_{X}Y_{0}, ĝ_{0}) yields

Θ(î∘ i)^{−1} ∘ Θ(id_{C_{X}Y_{0}} ∪_{X} î_{1})(θ(ĝ_{0} ∪_{g_{0}} g)) = Θ(î∘ i)^{−1} ∘ Θ(î)(θ(g))
+ θ(f) − Θ(î_{0} ∘ i_{0})^{−1} ∘ Θ(i_{0}))(θ(g_{0})).

Hence it remains to show

Θ(î∘ i))^{−1} ∘ Θ(id_{C_{X}Y_{0}} ∪_{X} î_{1})(θ(ĝ_{0} ∪_{g_{0}} g)) = Θ(î_{1} ∘ i_{1})^{−1} ∘ Θ(î_{1})(θ(p ∘ (ĝ_{0} ∪_{g_{0}} g) ∘ j)).

Applying to j : (Y_{1}, p ∘ ĝ_{0} ∪_{g_{0}} g ∘ j) ⟶ (C_{X}Y_{0} ∪_{X} Y_{1}, j ∘ p ∘ ĝ_{0} ∪_{g_{0}} g) invariance under homotopy equivalence yields

Θ(j)(θ(p ∘ ĝ_{0} ∪_{g_{0}} g ∘ j) = θ(j ∘ p ∘ ĝ_{0} ∪_{g_{0}} g).

Hence it suffices to show

Θ(î∘ i))^{−1} ∘ Θ(id_{C_{X}Y_{0}} ∪_{X} î_{1})(θ(ĝ_{0} ∪_{g_{0}} g)) = Θ(î∘ i))^{−1} ∘ Θ(id_{C_{X}Y_{0}} ∪_{X} î_{1})(θ(j ∘ p ∘ ĝ_{0} ∪_{g_{0}} g)).

This follows analogous to the proof that τ_{Y} is well-defined using the fact that p ∘ j is the identity and j ∘ p is homotopic relative X to the identity and hence (j ∘ p ∘ ĝ, j ∘ p ∘ ĝ_{0} ∪_{g_{0}} g) and (ĝ, ĝ_{0} ∪_{g_{0}} g) are homotopic relative X. This finishes the proof that τ_{d} in 4.4 of welldefined.

Next we show that τ_{d} is independent of d ≥ 2. If Y is a d-extension of X, then Σ_{X}Y is a d + 1-extension and there is a bijective correspondence between the d cells in Y − X and the d + 1-cells in Σ_{X}Y − X. In particular, the suspension of C(Ỹ, X̃) is C({ \atop Σ_{X}Y} , X̃). One easily checks that the following diagram commutes

[(C_{X}Y, Y), (C_{X}Y, Y)]_{f}  ⟶^{Σ_{X}} [(C_{X}Σ_{X}Y, Σ_{X}Y), (C_{X}Σ_{X}Y, Σ_{X}Y]_{f}
η↓  ↓^{η}
hom_{ℤπ_{1}(X,x)}(C_{d}(Ỹ, X̃), ϕ^{∗}C_{d}(Z̃, X̃)) ⟶^{Σ} (C_{d+1}({ \atop Σ_{X}Y} , X̃), ϕ^{∗}C_{d+1}({ \atop Σ_{X}Y} , X̃))

Hence it suffices to show

θ(Σ_{X}g) − θ(j ∘ i)(θ(f) = −Θ(j)(θ(g)) + θ(j ∘ i)(θ(f)  (4.5)

for a map (ĝ, g) : (C_{X}Y, Y) ⟶ (C_{X}Y, Y) with g|_{X} = f where j : Y ⟶ Σ_{X}Y is the canonical inclusion. Since Σ_{X}g is defined as ĝ ∪_{g} ĝ and î ∘ i is a homotopy equivalence, this follows from additivity and homotopy invariance under homotopy equivalences. This finishes the proof of τ_{d+1} = τ_{d} for d ≥ 2. In the sequel we write

τ := = τ_{d}  for d ≥ 2.  (4.6)

We have defined a map

τ(X, f) : U(X, f) ⟶ Θ(X, f)

for an endomorphism f : X ⟶ X of a finite connected CW -complex X. If X is not connected, we always assume π_{0}(f) = id and we define τ(X, f) by requiring that the following diagram commutes whose horizontal arrows are induced by the various inclusions of the components of X and whose upper horizontal arrow is an isomorphism

⊕_{C∈π_{0}(X)}U(C, f|_{C}) ⟶^{≅} U(X, f)
⊕_{C∈π_{0}(X)}τ(C,f|_{C})↓  ↓^{τ(X,f)}
⊕_{C∈π_{0}(X)}Θ(C, f|_{C}) ⟶ Θ(X, f)

Next we show that τ defines a natural transformation, i.e. we must show for any morphism h : (X_{0}, f_{0}) ⟶ (X_{1}, f_{1}) in End(C) that τ(X_{1}, f_{1}) ∘ U(h) = Θ(h) ∘ τ(X_{0}, f_{0}). It suffices to prove this in the special case where X_{0} and X_{1} are connected. Let Y_{0} be a d-extension of X for some d ≥ 2 and (ĝ_{0}, g_{0}) : (C_{X_{0}}Y_{0}, Y_{0}) ⟶ (C_{X_{0}}Y_{0}, Y_{0}) be an endomorphism with g|_{X} = f_{0}. Let Y_{1} be the retractive space over X given by the push out of Y_{0} ⟵^{i_{0}} X_{0} ⟶^{h} X_{1} and let (ĝ_{1}, g_{1}) : (C_{X_{1}}Y_{1}, Y_{1}) ⟶ (C_{X_{1}}Y_{1}, Y_{1}) be given by the push out property and g_{0}, f_{0} and f_{1}. Then the induction of the ℤπ_{1}(X_{0}, x_{0})-chain map C(g̃_{0}, f̃_{0}) : C(Ỹ_{0}, X̃_{0}) ⟶ ϕ_{0}^{∗}C(Ỹ_{0}, X̃_{0}) with the map induced by π_{1}(h, x) : π_{1}(x_{0}, x_{0}) ⟶ π_{1}(X_{1}, h(x_{0})) is the ℤπ_{1}(X_{1}, h(x_{0}))-chain map C(g̃_{1}, f̃_{1}) : C(Ỹ_{1}, X̃_{1}) ⟶ ϕ_{1}^{∗}C(Ỹ_{1}, X̃_{1}) where we have fixed a base point x_{0} ∈ X_{0} and a path w from f_{0}(x_{0}) to x_{0} and use for X_{1} the base point h(x_{0}) and the path h(w). Hence it suffices to show

Θ(h) ∘ Θ(î_{0} ∘ i_{0})^{−1} ∘ Θ(i_{0})(θ(g_{0})) − Θ(h)(Θ(f_{0}) = Θ(î_{1} ∘ i_{1})^{−1} ∘ Θ(i_{1})(θ(g_{1})) − θ(f_{1}).

This follows from additivity applied to g_{1} = g_{0} ∪_{f_{0}} f_{1}. This finishes the definition of the natural transformation

τ : U ⟶ Θ  (4.7)

between functors from End(C) to ABEL.

Next we have to show that τ(X, f) maps u(X, f) to θ(X, f) and that the natural transformation τ is uniquely determined by this property. Let Y_{n} be the push out

X_{n−1} ⟶^{j_{n−1}} X
k_{n−1}↓  ↓^{i_{n}}
X_{n} ⟶^{l_{n}} Y_{n}

where all arrows are canonical inclusions. There is a canonical retraction r_{n} : Y_{n} ⟶ X induced by the inclusions of X_{n−1}, X_{n} and X in X. If f_{k} : X_{k} ⟶ X_{k} is the restriction of f to X_{k}, then the push out above yields an endomorphism g_{n} : Y_{n} ⟶ Y_{n} defined by g_{n} = f_{n} ∪_{f_{n−1}} f. We obtain in Θ(Y_{n}) from additivity

θ(g_{n}) = Θ(i_{n})(θ(f)) + Θ(l_{n})(θ(f_{n})) − Θ(l_{n} ∘ k_{n−1})(θ(f_{n−1})).

Applying Θ(r_{n}) yields

Θ(r_{n})(θ(g_{n})) = θ(f) + Θ(j_{n})(θ(f_{n})) − Θ(j_{n−1})(θ(f_{n−1})).

We conclude in Θ(X, f):

θ(f) = ∑_{n=0}^{dim(Y)} (Θ(r_{n})(θ(g_{n})) − θ(f)) .  (4.8)

In particular we get for (U, u) in U(X, f)

u(f) = ∑_{n=0}^{dim(Y)} (U(r_{n})(u(g_{n})) − u(f)) .  (4.9)

If n ≥ 2, then Y_{n} is a n-extension of X for n ≥ 2 and because of Theorem 1.4

U(r_{n})(u(g_{n})) − u(f) = u(C_{n}(g̃_{n}, f̃)).

Since r_{n} ∘ g_{n} = f ∘ r_{n} holds, there is a canonical extension ĝ_{n} of g_{n}. Since î∘ i ∘ r_{n} and î are homotopic morphisms from (Y, g_{n}) to (C_{X}, ĝ_{n}) in End(C) we conclude from homotopy invariance Θ(r_{n}) = Θ(î∘ i)^{−1} ∘ Θ(i) . Hence the definition of τ(X, f) implies for n ≥ 2

τ(X, f) (U(r_{n})(u(g_{n})) − u(f)) = Θ(r_{n})(θ(g_{n})) − θ(f).  (4.10)

Next we show that equation 4.10 holds also for n = 0, 1. The following two equations are direct consequences of 4.5 (which is true for all n ≥ 0)

Θ(r_{n})(θ(g_{n})) − θ(f) = −Θ(Σ_{X}r_{n})(θ(Σ_{X}g_{n})) + θ(f);
τ(U(r_{n})(u(g_{n})) − u(f)) = −τ(U(Σ_{X}r_{n})(u(Σ_{X}g_{n})) + u(f).

Notice that Σ_{X}Y is a n + 1-extension if Y is an n-extension and Σ_{X}r_{n} plays the role of r_{n+1} for Σ_{X}Y. Hence 4.10 holds for n if it is true for n + 1. Therefore 4.10 is true for all n ≥ 0. Now 4.8, 4.9 and 4.10 imply

τ(X, f)(u(X, f)) = θ(X, f).  (4.11)

Notice that τ is uniquely determined by property 4.11 since any element in U(X, f) can be realized for an d-extension Y for d ≥ 2 and endomorphism (ĝ, g) : (C_{X}Y, Y) ⟶ (C_{X}Y, Y) by

U(î∘ i)^{−1} ∘ U(î)(u(g)) − u(f).

This finishes the proof of Theorem 2.5.

Remark 4.12. One may think that one could also use the following easier construction instead of the map τ_{Y}. Namely, given an endomorphism f : X ⟶ X and a retractive space Y over X one may consider endomorphisms g : Y ⟶ Y satisfying r ∘ g = f ∘ r. If [Y, Y]_{f}^{r} is the set of homotopy classes of such maps where the homotopy h also satisfies r ∘ h_{t} = f ∘ r for all t ∈ [0, 1], one can define a map

Θ′_{Y,f} : [Y, Y]_{f}^{r} ⟶ Θ(X, f)

by sending [g] to Θ(r)(θ(g)). There is an obvious map

e : [Y, Y]_{f}^{r} ⟶ [(C_{X}Y, Y), C_{X}Y)]_{f}

satisfying τ_{Y} ∘ e = τ′_{Y} since any map g : Y ⟶ Y with r ∘ g = f ∘ r has a canonical extension ĝ: C_{X}Y ⟶ C_{X}Y and Θ(î∘ i)^{−1} ∘ Θ(i) = Θ(r) holds. The problem is, however, that e is not bijective and hence the passage from geometry to algebra in Lemma 4.2 does not work for [Y, Y]_{f}^{r}. That e is not bijective, can be easily seen from the example Y = X ∨ S^{2}. Then a map g : Y ⟶ Y satisfying r ∘ g = f ∘ r is the same as a pointed map S^{2} ⟶ S^{2} and [Y, Y]_{f}^{r} is isomorphic to ℤ whereas [(C_{X}Y, Y), C_{X}Y)]_{f} is in general larger, namely hom_{ℤπ}(ℤπ, ϕ^{∗}ℤπ).

5 The construction of the transfer map

Suppose we are given the following geometric data:

Data 5.1. Let F ⟶ E ⟶^{p} B be a fibration of spaces of the homotopy type of connected finite CW -complexes and let

\begin{matrix}E & ⟶^{\overline{f}} & E \\ ^{p}↓ & & ↓^{p} \\ B & ⟶^{f} & B \end{matrix}

be a commutative square.

In this section we want to assign to these data a (natural) homomorphism

trf_{\overline{f},f} : U(f) ⟶ U(\overline{f}).  (5.2)

such that trf_{\overline{f},f}(u(f)) = u(\overline{f}) holds and examine its properties in Section 6. Since our invariant u(f) determines other invariants as explained in Section 3 these results for u give also information for the other invariants. Transfer questions for fixed point theory have also been investigated for instance in [9], [10], and [18, Chapter IV].

Remark 5.3. We mention that (U(f), u(f)) also makes sense for endomorphisms f : X ⟶ X of spaces X of the homotopy type of a finite CW -complex. The algebraic definition of u(f : C ⟶ C) via chain complexes extends to chain complexes of the homotopy type of a finite chain complex using commutativity and homotopy invariance, namely, choose any chain homotopy equivalence g : C ⟶ D for a finite chain complex D and define u(f) by u(g ∘ f ∘ g^{−1}) for any chain homotopy inverse g^{−1} of g.

From the algebraic point of view we will need the following algebraic data to define the transfer.

Data 5.4. 1. A short exact sequence of groups {1} ⟶Δ ⟶^{i}Γ ⟶^{p}π ⟶{1};

2. A commutative diagram of group homomorphisms

{1} ⟶ Δ ⟶^{i} Γ ⟶^{p} π ⟶ {1}
\overline{ϕ}_{f} ↓ \overline{ϕ} ↓ ϕ ↓
{1} ⟶ Δ ⟶^{i} Γ ⟶^{p} π ⟶ {1}

3. A finite free ℤΔ-chain complex C together with a Γ-twist L in the sense of [27, Definition 5.1 on page 155]. (A Γ-twist L on C is a collection {[L(γ)] | γ ∈ Γ} of ℤΔhomotopy classes of ℤΔ-chain maps L(γ) : C ⟶ c_{γ}^{∗}C such that for δ ∈ Δ the class [L(δ)] is represented by the map l(δ) given by left multiplication with δ and that the ℤΔ-chain maps L(γ_{1})∘L(γ_{2}) and L(γ_{1}·γ_{2}) from C to (c_{γ_{1}·γ_{2}})^{∗}C are ℤΔ-chain homotopic where c_{γ} : Δ ⟶ Δ sends δ to γδγ^{−1});

4. A ℤΔ-chain homotopy class [t] of ℤΔ-chain maps t : C ⟶ \overline{ϕ}_{f}^{∗}C such that for any γ ∈ Γ the following diagram commutes up to ℤΔ-chain homotopy:

C ⟶^{t}  \overline{ϕ}_{f}^{∗}C
L(γ)↓  ↓^{\overline{ϕ}_{f}^{∗}L(\overline{ϕ}(γ))}
(c_{γ})^{∗}C ⟶^{(c_{γ})^{∗}t} (c_{\overline{ϕ}(γ)})^{∗}\overline{ϕ}^{∗}C = \overline{ϕ}_{f}^{∗}(c_{γ})^{∗}C

One can think of a Γ-twist as an extension of the Δ-operation to a Γ-operation up to homotopy.

Example 5.5. Next we explain how the geometric data 5.1 yield algebraic data 5.4. Choose a point e ∈ E and a path w from \overline{f}(e) to e. Put

Δ = ker (π_{1}(p, e) : π_{1}(E, e) ⟶ π_{1}(B, p(e))) ;
Γ = π_{1}(E, e);
π = π_{1}(B, p(e)).

Then we obtain a commutative square with exact sequence rows

{1} ⟶ Δ ⟶^{i} Γ ⟶^{p} π ⟶ {1}
\overline{ϕ}_{f} ↓ \overline{ϕ} ↓ ϕ ↓
{1} ⟶ Δ ⟶^{i} Γ ⟶^{p} π ⟶ {1}

where i is given by the inclusion, p is π_{1}(p, e) and \overline{ϕ} is given by

\overline{ϕ} : Γ = π_{1}(E, e) ⟶^{π_{1}(\overline{f},e)} π_{1}(E, f(e)) ⟶^{c_{w}} Γπ_{1}(E, e).

Any path u from e_{0} to e_{1} in E defines a pointed homotopy class of pointed maps depending only on the homotopy class of u relative endpoints, the so called pointed fibre transport (see [26, section 6])

σ(u) : (F_{e_{1}}, e_{1}) ⟶ (F_{e_{0}}, e_{0}).

We have σ(u ∗ v) = σ(u) ∘ σ(v) and σ applied to the trivial path is represented by the identity. In particular we get for each e ∈ E a homomorphism

σ_{e} : π_{1}(E, e) ⟶ [(F_{e}, e), (F_{e}, e)]^{+}

from the fundamental group into the monoid of pointed homotopy classes of pointed selfmaps of (F_{e}, e). If s(u) : (F_{e}, e) ⟶ (F_{e}, e) is a representative of σ(u) and q : (\overline{F_{e}}, \overline{e}) ⟶ (F_{e}, e) is the covering of F_{e} associated to the epimorphism π_{1}(F_{e}, e) ⟶ Δ induced by the inclusion of F_{e} into E, then there is a unique lift \overline{s(u)} : (\overline{F_{e}}, \overline{e}) ⟶ (\overline{F_{e}}, \overline{e}). This map is c_{u} : Δ_{e} ⟶ Δ_{e}- equivariant where as before c_{u} is conjugation with u. Its c_{u}-homotopy class depends only on u and not on the choice of s(u) ∈ σ(u). Thus we obtain a Γ = π_{1}(E, e)-twist on the ℤΔ-chain complex C(\overline{F_{e}}) by

L(u) = [C(s(u))].

Recall that we have chosen a path w from \overline{f}(e) to e. Let s(w^{−}) : (F_{e}, \overline{f}(e)) ⟶ (F_{e}, e) be a representative of σ(w^{−}) where w^{−} is the inverse of w. Then s(w) ∘ \overline{f} |_{F_{e}}: (F_{e}, e) ⟶ (F_{e}, e) lifts uniquely to a map

\overline{s(w) ∘ \overline{f} |_{F_{e}}} : (\overline{F_{e}}, \overline{e}) ⟶ (\overline{F_{e}}, \overline{e}).

This map is \overline{ϕ}-equivariant and its equivariant homotopy class depends only on the homotopy class relative end points of w. Define a ℤΔ-chain map

t = C(\overline{s(w^{−}) ∘ \overline{f} |_{F_{e}}}) : C(\overline{F_{e}}) ⟶ \overline{ϕ}^{∗}C(\overline{F_{e}}).

Now we have the data 5.4.

For the data 5.4 we want to define a map

trf : U(π, ϕ) ⟶ U(Γ, \overline{ϕ}).  (5.6)

We will leave it to the reader to check that, given the data 5.1, the collection of the maps 5.6, which are obtained from the data 5.4 for the various choices of e ∈ E and path w from \overline{f}(e) to e as explained in Example 5.5, fit together to give the desired transfer map 5.2.

Consider a ℤπ-homomorphism α : ℤπ ⟶ ϕ^{∗}ℤπ. We can write α(1) = ∑_{w∈π} λ_{w} · w. Choose for any w ∈ π a lift \overline{w} ∈ Γ and for each γ ∈ Γ a representative L(γ) of [L(γ)]. Define a ℤΓ-chain map

X(α) : ℤΓ ⊗_{ℤΔ} C ⟶ \overline{ϕ}^{∗}ℤΓ ⊗_{ℤΔ} C γ ⊗ v ↦ ∑_{w∈π} λ_{w} · \overline{ϕ}(γ)\overline{w} ⊗ L(\overline{w}^{−1}) ∘ t(v). (5.7)

In order to check that this is a well-defined ℤΓ-chain map, we have to check that γ ⊗ v and γδ ⊗ δ^{−1}v have the same image

\overline{ϕ}(γδ)\overline{w}⊗L(\overline{w}^{−1})t(δ^{−1}v) = \overline{ϕ}(γ)\overline{w}\overline{w}^{−1}\overline{ϕ}(δ)\overline{w}⊗\overline{w}^{−1}\overline{ϕ}(δ^{−1})\overline{w}L(\overline{w}^{−1})t(v) = \overline{ϕ}(γ)\overline{w}⊗L(\overline{w}^{−1})t(v).

One easily checks that the ℤΓ-chain homotopy class of X(α) is independent of the choices of \overline{w} and L(γ). Given a ℤπ-homomorphism

α = (α_{i,j})_{i,j} : ⊕_{i=1}^{n}ℤπ ⟶ ⊕_{j=1}^{m}ℤπ

we obtain a ℤΓ-chain map unique up to ℤΓ-homotopy by

X(α) = (X(α_{i,j}))_{i,j} : ⊕_{i=1}^{n}ℤΓ ⊗_{ℤΔ} C ⟶ ⊕_{j=1}^{m}ℤΓ ⊗_{ℤΔ} C

We want to define the transfer map trf of 5.6 by requiring

trf ([α : ⊕_{i=1}^{n}ℤπ ⟶ ⊕_{i=1}^{n}ϕ^{∗}ℤπ]) = u (X(α) : ⊕_{i=1}^{n}ℤΓ ⊗_{ℤΔ} C ⟶ ⊕_{i=1}^{n}ϕ^{∗}ℤΓ ⊗_{ℤΔ} C) ,

where u is the invariant of Definition 1.2. Since the ℤΓ-chain homotopy class of X(α) depends only on α, the expression u(X(α)) is well-defined. In order to check that the map trf is well-defined one must verify that the relations in U(ℤπ, ϕ) are respected. Suppose we have a commutative square of ℤπ-maps with isomorphisms as vertical maps:

⊕_{i=1}^{n}ℤπ ⟶^{α} ⊕_{i=1}^{n}ϕ^{∗}ℤπ
β↓  ↓^{ϕ^{∗}β}
⊕_{i=1}^{n}ℤπ ⟶^{α′} ⊕_{i=1}^{n}ϕ^{∗}ℤπ

Then one can construct a ℤΓ-chain map

X(β) : ⊕_{i=1}^{n}ℤΓ ⊗_{ℤΔ} C ⟶ ⊕_{i=1}^{n}ℤΓ ⊗_{ℤΔ} C

analogously as for α using id instead of t. Then the following square commutes up to ℤΓchain homotopy:

⊕_{i=1}^{n}ℤΓ ⊗_{ℤΔ} C ⟶^{X(α)} ⊕_{i=1}^{n}\overline{ϕ}^{∗}ℤΓ ⊗_{ℤΔ} C
X(β)↓  ↓^{\overline{ϕ}^{∗}X(β)}
⊕_{i=1}^{n}ℤΓ ⊗_{ℤΔ} C ⟶^{X(α′)} ⊕_{i=1}^{n}\overline{ϕ}^{∗}ℤΓ ⊗_{ℤΔ} C

Since X(β^{−1}) is a ℤΓ-chain homotopy inverse of X(β), we get from homotopy invariance and Lemma 1.6

u(X(α)) = u(X(α′)).

Now consider a block endomorphism

α = ( \begin{matrix}α_{1} & α_{0} \\ 0 & α_{2} \end{matrix} ) : (⊕_{i=1}^{n}ℤπ) ⊕ (⊕_{j=1}^{m}ℤπ) ⟶ (⊕_{i=1}^{n}ℤπ) ⊕ (⊕_{j=1}^{m}ℤπ) .

Then X(α) also has block form and we get:

u(X(α)) = u(X(α_{1})) + u(X(α_{2})).

This finishes the construction of the transfer maps 5.2 and 5.6 and the proof that they are well-defined. The main result of this section is the next theroem. One should compare its proof with the corresponding identifications of algebraic and geometric transfers in algebraic K and L-theory given by finiteness obstructions, Whitehead torsion [26] and surgery obstructions [32].

Theorem 5.8. Given the geometric data 5.1, we get:

trf_{f,\overline{f}}(u(f)) = u(\overline{f}).

Proof. For each n ≥ 0 we define (Y_{n}, g_{n}) to be the push out in End(C)

(B_{n−1}, f_{n−1}) ⟶^{j_{n}} (B, f)
k_{n−1}↓  ↓^{i_{n}}
(B_{n}, f_{n}) ⟶^{l_{n}} (Y_{n}, g_{n})

The identity on B induces a retraction r_{n} : Y_{n} ⟶ B. By pulling back p : (E, \overline{f}) ⟶ (B, f) with r_{n} and restricting to B_{n−1}, B_{n} and B again we obtain a push out

(E |_{B_{n−1}}, \overline{f_{n−1}}) ⟶^{\overline{j_{n−1}}} (E, \overline{f})
\overline{k_{n−1}}↓  \sqrt{i_{n}}
(E |_{B_{n}}, \overline{f_{n}}) ⟶^{\overline{l_{n}}} (E_{n}, \overline{g_{n}})

and a retraction \overline{r_{n}} : (E_{n}, \overline{g_{n}}) ⟶ (E, \overline{f}) covering r_{n} : (Y_{n}, g_{n}) ⟶ (B, f). We obtain from additivity

u(B, f) = ∑_{n=0}^{dim(B)} (U(r_{n})(u(Y_{n}, g_{n})) − u(B, f)) ;

u(E, \overline{f}) = ∑_{n=0}^{dim(B)} (U(\overline{r_{n}})(u(E_{n}, \overline{g_{n}})) − u(E, \overline{f})) .

Hence it suffices to show for 0 ≤ n

trf_{f,\overline{f}} (U(r_{n})(u(Y_{n}, g_{n})) − u(B, f)) = U(\overline{r_{n}})(u(E_{n}, \overline{g_{n}})) − u(E, \overline{f}).  (5.9)

Next we show analogously as in the corresponding part of the proof of Theorem 2.5 that 5.9 holds for all n ≥ 0 if we can prove it for n ≥ 2. Since r_{n} : (Y_{n}, g_{n}) ⟶ (B, f) is a morphism in End(C), we can canonically extend r_{n} to a retraction r̂_{n} : C_{B}Y_{n} ⟶ B and g_{n} to ĝ_{n} : C_{B}Y_{n} ⟶ C_{B}Y_{n} such that r̂_{n} : (C_{B}Y_{n}, ĝ_{n}) ⟶ (B, f) is also a morphism in End(C). Define (Y_{nΣ}, g_{nΣ}) to be the push out in End(C)

(Y_{n}, g_{n}) ⟶^{i} (C_{B}Y_{n}, ĝ_{n})
i↓  ↓
(C_{B}Y_{n}, ĝ_{n}) ⟶ (Y_{nΣ}, g_{nΣ})

and define the retraction r_{nΣ} : (Y_{nΣ}, g_{nΣ}) ⟶ (B, f) by the push out property using r̂_{n} and r_{n}. By pulling back p : (E, \overline{f}) ⟶ (B, f) we obtain a push out in End(C)

(E_{n}, \overline{g_{n}}) ⟶ (Ê_{n}, \overline{ĝ_{n}})
↓  ↓
(Ê_{n}, \overline{ĝ_{n}}) ⟶ (E_{nΣ}, \overline{g_{nΣ}})

covering the previous push out and a retraction \overline{r_{nΣ}} : (E_{nΣ}, \overline{g_{nΣ}}) ⟶ (E, \overline{f}) covering r_{nΣ}. Now we get from additivity and invariance under homotopy equivalence and the fact that the inclusions B ⟶ Ŷ and E ⟶ Ê_{n} are homotopy equivalences

\begin{aligned}U(\overline{r_{nΣ}})(u(E_{nΣ}, \overline{g_{nΣ}})) & = & −U(\overline{r_{n}})(u(E, \overline{g_{n}})) & ∈ U(E, \overline{f}); \\ U(r_{nΣ})(u(Y_{nΣ}, g_{nΣ})) & = & −U(r_{n})(u(Y_{n}, g_{n})) & ∈ U(B, f). \end{aligned}

Hence

trf_{f,\overline{f}} (U(r_{n})(u(Y, g_{n}))) = u(E, \overline{g_{n}})

is equivalent to

trf_{f,\overline{f}} (u(Ŷ_{nΣ}, g_{nΣ})) = u(E_{nΣ}, \overline{g_{nΣ}}).

Notice that Y_{nΣ} is a (n + 1)-extension of B. Thus we have reduced the proof of Theorem 5.8 to the proof of the following statement:

Let n ≥ 2 and r : (Y, g) ⟶ (B, f) be a retraction in End(C) such that Y is a n-extension of X. Define a retraction \overline{r} : (r^{∗}E, \overline{g}) ⟶ (E, \overline{f}) covering r by pulling back p : (E, \overline{f}) ⟶ (B, f). Then we get in U(E, f)

trf_{f,\overline{f}} (U(r)(u(Y, g)) − u(B, f)) = U(\overline{r})(u(r^{∗}E, \overline{g})) − u(E, \overline{f}).  (5.10)

Fix e ∈ E and a path w from \overline{f}(e) to e. Choose characteristic maps for the cells in Y − X:

∐_{i∈I} S^{n−1} \underrightarrow{∐_{i∈I} q_{i}} B
↓  ↓
∐_{i∈I} D^{n} \underrightarrow{∐_{i∈I} Q_{i}} Y

and paths w_{i} from q_{i}(1) to p(e) ∈ B. Using these path each cell defines an element b_{i} in π_{n}(Y, b, p(e)) such that {b_{i} | i ∈ I} is a ℤπ_{1}(B, p(e))-basis. With respect to this basis we can write C(g̃, f̃) : C(Ỹ, X̃) ⟶ ϕ^{∗}C(Ỹ, X̃) using the identification of C(Ỹ, X̃) with π_{n}(Y, B, p(e)) by the Hurewicz isomorphism as a ℤπ_{1}(B, p(e))-map

C(g̃, f̃) : ⊕_{i∈I}ℤπ_{1}(B, p(e)) ⟶ ϕ^{∗} ⊕_{i∈I} ℤπ_{1}(B, p(e)).

Notice that u(C(g̃, f̃)) ∈ U(ℤπ_{1}(B, p(e)), ϕ) represents U(r)(u(Y, g)) − u(B, f) ∈ U(B, f)). Using the paths w_{i} above there are up to strong fibre homotopy equivalence unique fibre trivilizations

T_{i} : π_{1}(E, e) ×_{Δ(e)} F̃_{e} × (D^{n}, S^{n−1}) ⟶ (Q̃_{i}^{∗}E, q̃_{i}^{∗}E).

They induce an explicit isomorphism

ρ : ⊕_{i∈I}ℤπ_{1}(E, e) ⊗_{Δ(e)} C(\overline{F_{e}}) ⟶ C(r̃^{∗}E, Ẽ).

Let

X(C(g̃, f̃)) : ⊕_{i∈I}ℤπ_{1}(E, e) ⊗_{Δ(e)} C(\overline{F_{e}}) ⟶ ⊕_{i∈I} \overline{ϕ}^{∗}ℤπ_{1}(E, e) ⊗_{Δ(e)} C(\overline{F_{e}})

be the ℤπ_{1}(E, e)-chain map unique up to ℤπ_{1}(E, e)-chain homotopy defined in section 7 for the choice of data 5.4 given by p : E(\overline{f}) ⟶ (B, f). Then we have by definition of the transfer that u(X(C(g̃, f̃))) ∈ U(ℤπ_{1}(E, e), \overline{ϕ}) represents trf_{f,\overline{f}} (U(r)(u(Y, g)) − u(B, f)). Notice that

u(C(˜\overline{g}, ˜ \overline{f}) : C(r̃^{∗}E, Ẽ) ⟶ C(r̃^{∗}E, Ẽ) ∈ U(ℤπ_{1}(E, e), \overline{ϕ})

represents U(\overline{r}) (u(r^{∗}E, \overline{g})) − u(E, \overline{f}) ∈ U(E, \overline{f})). Hence we have to show

u (X(C(g̃, f̃))) = u (C(˜\overline{g}, ˜ \overline{f})) ∈ U(ℤπ_{1}(E, e), \overline{ϕ}).

This claim follows if the following diagram of ℤπ_{1}(E, e)-chain complexes commutes up to ℤπ_{1}(E, e)-chain homotopy

⊕_{i∈I}ℤπ_{1}(E, e) ⊗_{Δ(e)} C(\overline{F_{e}}) ⟶^{X(C(g̃,f̃))} ⊕_{i∈I}ℤπ_{1}(E, e) ⊗_{Δ(e)} C(\overline{F_{e}})
ρ↓  ↓^{ρ}
C(r̃^{∗}E, Ẽ)  ⟶^{C(˜\overline{g},˜\overline{f})}  C(r̃^{∗}E, Ẽ)

The proof is omitted since it is a straightforward modification of the proof of Theorem 2.2 in [26, section 7]. This finishes the proof of Theorem 5.8.

Remark 5.11. We mention that one can analogously define a transfer map

trf_{\overline{f},f} : Λ(f) ⟶ Λ(\overline{f})  (5.12)

satisfying

trf_{f,\overline{f}}(λ(f)) = λ(\overline{f})

for the functorial Lefschetz invariant (Λ, λ) of Example 3.5. It sends the class [u] ∈ Λ(ℤπ, Φ) to tr_{(ℤΓ,\overline{Φ})}(X(r_{u})), where tr_{(ℤΓ,\overline{Φ})} has been defined in 3.6, r_{u} : ℤπ ⟶ Φ^{∗}ℤπ maps v to Φ(v)u and X(r_{u}) has been defined in 5.7.

6 Properties of the transfer map

In this section we prove vanishing results and a down-up-formula for the geometric transfer map. Moreover, we consider the special case S^{n} as fiber and treat (not necessarily free) S^{1}- actions. There are analogous computations of the transfer maps in algebraic K and L-theory concerning finiteness obstructions, Whitehead torsion [27] and surgery obstructions [33].

Theorem 6.1. Suppose we are given the data 5.1. Then the transfer map

trf_{\overline{f},f} : U(B, f) ⟶ U(E, \overline{f})

and the element

u(E, \overline{f}) ∈ U(E, \overline{f})

vanish if one of the following conditions are satisfied:

1. p is untwisted, i.e. the pointed fibre transport σ_{e} : π_{1}(E, e) ⟶ [(F_{p(e)}, e), (F_{p(e)}, e)]^{+} (see [26, section 6])) is trivial. The fundamental group π_{1}(F) of F is non-trivial or the Euler characteristic of F satisfies χ(F) = 0. Moreover, the composition of \overline{f} |_{F_{b}}: F_{b} ⟶ F_{f(b)} and ω(w) : F_{f(b)} ⟶ F_{b} is homotopic to the identity on F_{b} for some (and hence all) b ∈ B and path w from b to f(b), where ω(w) is given by the (free) fibre transport along paths in the base space;

2. The map π_{1}(F, e) ⟶ π_{1}(E, e) induced by the inclusion is trivial and π_{1}(B) operates trivially on H_{p}(F) for all p ≥ 0. The fundamental group π_{1}(F) of F is non-trivial or the Euler characteristic of F satisfies χ(F) = 0. Moreover, the map induced by the composition of \overline{f} |_{F_{b}}: F_{b} ⟶ F_{f(b)} and ω(w) : F_{f(b)} ⟶ F_{b} on H_{∗}(F_{b}) is trivial for some (and hence all) b ∈ B and path w from b to f(b).

This theorem is a direct consequence of [27, section 4], Lemma 6.2 and Lemma 6.5 which we will prove below.

Lemma 6.2. Assume for the data 5.4 that Δ is central in Γ, the ℤΔ-chain map l(δ) : C ⟶ C given by multiplication with δ is ℤΔ-chain homotopic to id and the Γ-twist L on C is trivial, i.e., for all γ ∈ Γ we have [L(γ)] = [id]. Assume furthermore \overline{ϕ}_{f} = id and t = id. Then trf is zero if Δ is non-trivial. If Δ is trivial and we identify Γ and π, the transfer tr is given by multiplication with the Euler characteristic χ(C) = ∑_{i≥0}(−1)^{i} · dim_{ℤΔ}(C_{i}).

Proof. We get for α : ⊕_{i=1}^{n}ℤπ ⟶ ⊕_{j=1}^{n}ϕ^{∗}ℤπ

trf(u(α)) = ∑_{i≥0}(−1)^{i} · u ((X(α)_{i} : ℤΓ ⊗_{ℤΔ} C_{i} ⟶ ℤΓ ⊗_{ℤΔ} C_{i})

One easily checks that for an appropriate ℤΓ-map β : ⊕_{k=1}^{n}ℤΓ ⟶ ⊕_{k=1}^{n}\overline{ϕ}^{∗}ℤΓ we have for all i ≥ 0

X(α)_{i} = ⊕_{j=1}^{dim_{ℤΔ}(C_{i})}β.

This implies

trf(u(α)) = χ(C) · u(β).

If Δ is trivial, β can be chosen to be α if we identify Γ and π. Suppose that Δ is nontrivial. Then the classical Lefschetz number, here with values in the commutative ring ℤΔ, of the ℤΔ-chain map l(δ) : C ⟶ C is χ(C) · δ ∈ ℤΔ for all δ ∈ Δ. Since l(d) and l(1) are ℤΔ-chain homotopic by assumption, we have for all δ ∈ Δ:

χ(C) · d = χ(C) · 1 ∈ ℤΔ.

This implies χ(C) = 0. This finishes the proof of Lemma 6.2.

Let p_{∗} : U(ℤΓ, \overline{ϕ}) ⟶ U(ℤπ, ϕ) be the map induced by induction with p : ℤΓ ⟶ ℤπ. Notice that this map is well-defined since we require ϕ ∘ p = p ∘ \overline{ϕ}. Let Sw^{f}(π, ϕ) respectively Sw(π, ϕ) be the Grothendieck group of ℤπ-maps f : M ⟶ Φ^{∗}M for ℤπ-modules M which are finitely generated free respectively finitely generated over ℤ. In other words, its definition is analogous to the definition of U(ℤπ, ϕ) in Definition 1.2 with the exception that all ℤπmodules are required to be finitely generated free respectively finitely generated over ℤ instead of requiring that they are finitely generated free over ℤπ. The tensor product over ℤ with the diagonal π-action induces a pairing

⊗_{ℤ} : Sw^{f}(π, ϕ) ⊗ U(ℤπ, ϕ) ⟶ U(ℤπ, ϕ).  (6.3)

Notice that the analogous pairing for Sw(π, ϕ) is not well-defined because tensoring over ℤ with a finitely generated abelian group M is an exact functor if and only if M is free. Given the data 5.4 there is an element in Sw(π, ϕ)

h(C, L) := ∑_{i≥0}(−1)^{i} · [H_{i}(ℤ ⊗_{ℤΔ} t) : H_{i}(ℤ ⊗_{ℤΔ} C) ⟶ ϕ^{∗}H_{i}(ℤ ⊗_{ℤΔ} C)], (6.4)

where the action of w ∈ π on H_{i}(ℤ ⊗_{Δ} C) is induced by

ℤ ⊗_{Δ} C ⟶ ℤ ⊗_{Δ} C n ⊗ v ↦ n ⊗ L(\overline{w})(v)

for any lift \overline{w} ∈ Γ of w. The next lemma is a down-up formula for the transfer, i.e. it computes the composition of the transfer with the map induced by p.

Lemma 6.5.

1. The canonical map I : Sw^{f}(π, ϕ) ⟶ Sw(π, ϕ) is an isomorphism.

2. The composition p_{∗} ∘ trf : U(ℤπ, ϕ) ⟶ U(ℤπ, ϕ) is given by the pairing 6.3 and the element I^{−1}(h(C, L)) ∈ Sw^{f}(π, ϕ) for the element h(C, L) ∈ Sw(π, ϕ) defined in 6.4.

3. Assume that the π-action on H_{∗}(ℤ ⊗_{ℤΔ} C) is trivial and H_{i}(t) is the identity for all i ≥ 0. Then the composition p_{∗} ∘ trf : U(ℤπ, ϕ) ⟶ U(ℤπ, ϕ) is multiplication with the Euler characteristic χ(C) = ∑_{i≥0}(−1)^{i} · dim_{ℤ}(ℤ ⊗_{ℤΔ} C). If additionally Δ is trivial and χ(C) = 0, then trf vanishes.

Proof. This proof is a modification of the proof of [27, Corollary 6.4 on page 159] and [40, Lemma 2.2].

Corollary 6.6. Let G be a non-trivial connected compact Lie group and let X be a free G-CW -complex with compact quotient G\X. Let f : X ⟶ X be a G-map. Then

u(f) = 0. ∈ U(f).

Proof. This follows from Theorem 6.1.1 since G ⟶ X ⟶ G\X is an untwisted fibration over a finite CW -complex and χ(G) = 0.

We have the following result for S^{1}-actions.

Theorem 6.7. Let f : X ⟶ X be a S^{1}-endomorphism of a finite S^{1}-CW -complex X. Denote by i : (X^{S^{1}}, f^{S^{1}}) ⟶ (X, f) the morphism in End(C) induced by the inclusion of the fixed point set. Then we have:

U(i)(u(X^{S^{1}}, f^{S^{1}})) = u(X, f) ∈ U(X, f).

In particular u(X, f) vanishes if the S^{1}-action has no fixed points.

Proof. Since for any subgroup H ⊂ S^{1} different from S^{1} the quotient S^{1}/H is isomorphic to S^{1} again, one reduces the claim using additivity and induction over the orbit bundles and the skeletons to the following assertion: If Y is the S^{1}-push out

∐_{i∈I} S^{1} × S^{n−1} \underrightarrow{∐_{i∈I} q_{i}} X
↓  ↓^{i}
∐_{i∈I} S^{1} × D^{n} \underrightarrow{∐_{i∈I} Q_{i}} Y

for n ≥ 0 and (g, f) : (Y, X) ⟶ (Y, X) is a pair of S^{1}-maps, then we have

U(i)(u(X, f)) = u(Y, g)).

Notice that this would follow from Corollary 6.6 if S^{1} acts freely on X and we want to reduce the assertion to this case by the following construction. Choose a unitary S^{1}-representation V such that S^{1} acts freely on the unit sphere SV and dim(Y) < dim(SV) holds. Then SV × X with the diagonal S^{1}-action is free and the projection pr : SV × X ⟶ X is n + 1connected. Let k : S^{1} × S^{n−1} ⟶ SV be the composition of the projection onto S^{1} and an inclusion of an orbit. Define Z by the S^{1}-push out:

∐_{i∈I} S^{1} × S^{n−1} \underrightarrow{∐_{i∈I} k×q_{i}} SV × X
↓  ↓^{j}
∐_{i∈I} S^{1} × D^{n} \underrightarrow{∐_{i∈I} P_{i}}  Z

Let \overline{pr} : Z ⟶ Y be the map induced by the commutative diagram

∐_{i∈I} S^{1} × D^{n} ⟵ ∐_{i∈I} S^{1} × S^{n−1} \underrightarrow{∐_{i∈I} k×q_{i}} SV × X
id↓  ^{id}↓  ^{pr}↓
∐_{i∈I} S^{1} × D^{n} ⟵ ∐_{i∈I} S^{1} × S^{n−1} \underrightarrow{∐_{i∈I} q_{i}}  X

Then we obtain a S^{1}-push out

SV × X ⟶^{pr} X
j↓  ↓^{i}
Z ⟶^{\overline{pr}} Y

Let [Z, Z]_{id×f:SV ×X⟶SV ×X}^{S^{1}} be the S^{1}-equivariant homotopy classes of maps g : Z ⟶ Z relative id ×f : SV × X ⟶ SV × X. This means that such a map g extends id ×f and a S^{1}-homotopy connecting two such maps is stationary on SV × X. The push out property yields a map

l : [Z, Z]_{id×f:SV ×X⟶SV ×X}^{S^{1}} ⟶ [Y, Y]_{f:X⟶X}^{S^{1}}.  (6.8)

Next we want to show that this map is bijective. We obtain a bijection

[(Z, Z]_{id×f:SV ×X⟶SV ×X}^{S^{1}} ⟶ ∏_{i∈I}[D^{n}, Z]_{(id×f)∘(k×q_{i})|_{{1}×S^{n−1}}):S^{n−1}⟶SV ×X}

by sending (g, f) to (g ∘ P_{i} |_{{1}×D^{n}})_{i∈I}. The composition

\overline{pr} ∘ j ∘ (id ×f) ∘ (k × q_{i}) | \{1\} × S^{1} : S^{n−1} ⟶ Y

is nullhomotopic, a nullhomotopy comes from Q_{i} |_{{1}×D^{n}}. Since pr : SV × X ⟶ X and hence \overline{pr} is (n + 1)-connected we can extend j ∘ (id ×f) ∘ (k × q_{i}) |_{\{1\}×S^{n−1}}: S^{n−1} ⟶ Z to a map h_{i} : S_{−}^{n} ⟶ Z for i ∈ I where S_{−}^{n} denotes the lower hemisphere. There is an obvious bijection given by extending with h_{i} if we think of D^{n} as the upper hemisphere S_{+}^{n}:

[D^{n}, Z]_{(id×f)∘(k×q_{i})|_{{1}×S^{n−1}}:S^{n−1}⟶SV ×X} ⟶ [S^{n}, Z]_{h_{i}:S_{−}^{n}⟶Z}

The forgetful map, which remembers only that the base point s ∈ S^{n−1} ⊂∈ S_{−}^{n} is mapped to x_{i} := j ∘ (id ×f) ∘ (k × q_{i})(1, s), is a map

[S^{n}, Z]_{h_{i}:S_{−}^{n}⟶Z} ⟶ π_{n}(Z, x_{i}).

Since S_{−}^{n} is contractible in S^{n}, a cofibration argument shows that this map is bijective. Hence we have constructed a bijection

[Z, Z]_{id×f:SV ×X⟶SV ×X}^{S^{1}} ⟶ ∏_{i∈I} π_{n}(Z, x_{i})

depending on the choice of the extensions h_{i}. These extensions h_{i} induce by composition with \overline{pr} extensions of i ∘ f ∘ q_{i} |_{\{1\}×S^{n−1}}: S^{n−1} ⟶ Y. With respect to these choices we obtain analogously a bijection [Y, Y]_{f:X⟶X}^{S^{1}} ⟶ π_{n}(Y, pr(x_{i})) such that the following diagram commutes

[Z, Z]_{id×f:SV ×X⟶SV ×X}^{S^{1}} ⟶^{≅} ∏_{i∈I} π_{n}(Z, x_{i})
l↓  \sqrt{pr}
[Y, Y]_{f:X⟶X}^{S^{1}}  ⟶^{≅} ∏_{i∈I} π_{n}(Y, pr(x_{i}))

As SV is (n + 1)-connected, pr and hence \overline{pr} are (n + 1)-connected. Hence the map l in 6.8 is bijective.

We conclude that we can find for the given S^{1}-extension g : Y ⟶ Y of f : X ⟶ X an S^{1}-extension g′ : Z ⟶ Z and a homotopy h : Y × [0, 1] ⟶ Y relative f from l(g′) to g where l(g′) : Y ⟶ Y is given by the pushout property by g′ ∪_{id_{SV} ×f} f. We derive from additivity

U(\overline{pr}) (u(g′) − U(j)(u(id_{SV} ×f))) = u(l(g′)) − U(i)(u(f)) ∈ U(Y, l(g′)).

Since S^{1} acts freely on Z and SV × X, Theorem 6.1 implies u(g′) = 0 and u(id_{SV} ×f) = 0. This shows

u(l(g′)) − U(i)(u(f)) = 0 ∈ U(Y, l(g′)).

Because of invariance under homotopy and homotopy equivalence we conclude from the commutativity of the following diagram

U(X, f) ⟶^{i_{0}} U(X × [0, 1], id ×f) ⟵^{i_{1}} U(X, f)
U(i)↓  ^{U(i×id)}↓  ^{U(i)}↓
U(Y, l(g′)) ⟶^{i_{0}} U(Y × [0, 1], h × pr_{[0,1]}) ⟵^{i_{1}} U(Y, g)

that

U(i_{0}) (u(l(g′)) − U(i)(u(f))) = u(h×pr_{[0,1]})−U(i×id)(u(id ×f)) = U(i_{1})(u(g)−U(i)(u(f)))

holds. This implies

u(g) − U(i)(u(f)) = U(i_{1})^{−1}U(i_{0}) (u(l(g′)) − U(i)(u(f))) = 0 ∈ U(Y, g).

This finishes the proof of Theorem 6.7.

Example 6.9. We consider the special case of data 5.1, where F is the sphere S^{n} for n ≥ 1. Fix a base point b ∈ B and a path w from f(b) to b. Put π = π_{1}(B, b). Let d be the degree of the endomorphism of S_{b}^{n} given by the composition of \overline{f} |_{F_{b}}: F_{b} ⟶ F_{f(b)} and ω(w) : F_{f(b)} ⟶ F_{b} where ω(w) is given by the (free) fiber transport. Let ϵ : π ⟶ {±1} be the homomorphism which sends a path u to the degree of the map ω(u) : S_{b}^{n} ⟶ S_{b}^{n}. Then we get from Lemma 6.5 that p_{∗} ∘ trf : U(ℤπ, ϕ) ⟶ U(ℤπ, ϕ) is given by

p_{∗} ∘ trf ([f : M → ϕ^{∗}M]) = [f : M → ϕ^{∗}M] + (−1)^{n} · [d · f : M^{ϵ} → ϕ^{∗}(M^{ϵ})], (6.10)

where f : M ⟶ ϕ^{∗}M is a ℤπ-endomorphism for a finitely generated free ℤπ-module M and M^{ϵ} is the ℤπ-module which has the same underlying abelian group as M and has the new π-action given by u · x = ϵ(u)u · x for x ∈ M and u ∈ π.

If ϵ is trivial, d = 1 and n is odd, we conclude from 6.10 for n ≥ 3 and from Lemma 6.1.1 for n = 1 that trf is trivial. If ϵ is trivial, d = 1 and n is even, we get p_{∗} ∘ trf = 2 · id from 6.10 and p_{∗} is an isomorphism since π_{1}(p) is bijective.

We have introduced in Section 3 other functorial Lefschetz invariants and we will explain what the results for the universal invariant (U, u) implies for them. This is obvious if ϵ is trivial and d = 1 from the computation above. Let us consider in the sequel the case for arbitrary ϵ and d.

We begin with the generalized Lefschetz invariant of Example 3.5 for n ≥ 2. In the sequel we will identify π_{1}(E) and π by π_{1}(p) and in particular Λ(ℤπ_{1}(E), \overline{Φ}) with Λ(ℤπ, Φ). Define a map

trf′ : Λ(ℤπ, Φ) ⟶ Λ(ℤπ, Φ) [∑_{v∈π} λ_{v} · v] ↦ [∑_{v∈π} λ_{v} · v] − (−1)^{n} · d · [∑_{v∈π} λ_{v}ϵ(v) · v]

Then we get under the identifications above

λ(\overline{f}) = trf′(λ(f)).

Notice that even in the case where n is odd and d = 1 it can happen that the (classical Lefschetz number) λ_{class}(f) (see Example 3.1) vanishes, whereas λ_{class}(\overline{f}) is not zero. The reason is that for non-trivial ϵ there is no map trf′′ : ℤ ⟶ ℤ satisfying e ∘ trf′ = trf′′ ∘e, where e : Λ(ℤπ, Φ) ⟶ ℤ maps [∑_{v∈π} λ_{v} · v] to ∑_{v∈π} λ_{v}.

Next we consider the invariant L^{2}-torsion ρ^{(2)}(T_{f}) ∈ ℝ of Example 3.12. If d = 1 and n ≥ 1 we get for all possible ϵ the same result, namely,

ρ^{(2)}(T\overline{f}) = (1 + (−1)^{n}) · ρ^{(2)}(T_{f}).

If ϵ is trivial, this follows from the computation we have done already above. If ϵ is non-trivial, there is a two-sheeted covering q : E′ ⟶ E such that ϵ for the S^{n}- fibration p ∘ q : E′ ⟶ B is trivial. Now the general case follows from the fact that ρ^{(2)} is multiplicative under finite coverings [29, Theorem 1.10].

7 The mapping torus approach

In this section we use the construction of the mapping torus to reduce computations to the case where the ring endomorphism ϕ is the identity.

Let R be an associative ring with unit and let ϕ : R ⟶ R be a ring homomorphism respecting the unit. Denote by R̂ the ring given by the colimit of the direct system of rings indexed by the integers

… ⟶^{ϕ} R ⟶^{ϕ} R ⟶^{ϕ} R ⟶^{ϕ} …

A model is ℤ × R/ ∼ where ∼ is the equivalence relation for which r ∈ n × R and s ∈ m × R satisfy r ∼ s if and only if ϕ^{k−n}(r) = ϕ^{k−m}(s) holds for an appropriate integer k with k ≥ m, n. Let i : R ⟶ R̂ be the canonical homomorphism which sends r to the class of 0 × r. Notice that its kernel is the union ∪_{n≥1} ker(ϕ^{n}). Let ϕ̂ : R̂ ⟶ R̂ be the canonical automorphism which sends the class of n × r to (n − 1) × r. We have ϕ̂∘ i = i ∘ ϕ. The ring R̂[t, t^{−1}]ϕ̂ consists of formal finite Laurent series ∑_{n=k}^{l} t^{n}a_{n} for integers k and l and a_{n} ∈ R̂. Addition is given by

(∑_{n} t^{n}a_{n}) + (∑_{n} t^{n}b_{n}) := ∑_{n} t^{n}(a_{n} + b_{n})

and multiplication by

(∑_{n} t^{n}a_{n}) · (∑_{n} t^{n}b_{n}) := ∑_{n} t^{n} (∑_{k} ϕ̂^{k}(a_{n−k})b_{k}) .

Notice that multiplication is essentially given by at = tϕ̂(a) for a ∈ R̂. Next we want to construct a commutative square

7.1.  \begin{matrix}U(R, ϕ) & ⟶^{τ} & U(R̂[t, t^{−1}]ϕ̂, id) \\ ^{η}↓ & & ^{η}↓ \\ ∏_{m≥1} Λ(R, ϕ^{m}) & \underrightarrow{τ=∏_{m≥1} τ_{m}} & ∏_{m≥1} Λ(R̂[t, t^{−1}]ϕ̂, id) \end{matrix}

The group Λ(R, ϕ) is the obvious generalization of the special case of 3.7 where R is ℤπ and ϕ is induced by a group endomorphism of π, namely

Definition 7.2. Let H ⊂ R be the abelian subgroup of R generated by all elements of the form ϕ(r)s − sr. Define the abelian group

Λ(R, ϕ) := R/H.

Consider an R-map f : F ⟶ ϕ^{∗}F for a finitely generated free (left) R-module F. Choose a basis {b_{1}, …, b_{k}} for F. The (k, k)-matrix A with entries in R describing f is determined by the property f(b_{i}) = ∑_{j=1}^{k} A_{i,j}b_{j}. Define

tr_{(R,ϕ)}(f) := ∑_{i=1}^{k}[A_{i,i}] ∈ Λ(R, ϕ).  (7.3)

One easily checks that this definition is independent of the choice of the basis. The m-th component η_{m} of the map η is given on representatives by

η_{m}[f] := [tr_{(R,ϕ^{m})}(f^{m})].  (7.4)

Next we define the two horizontal maps. Let f : F ⟶ ϕ^{∗}F be an endomorphism of a finitely generated free (left) R-module F. Then we obtain a R̂[t, t^{−1}]ϕ̂-endomorphism of a finitely generated free R̂[t, t^{−1}]ϕ̂-module

r_{t} ⊗ f : R̂[t, t^{−1}]ϕ̂ ⊗_{R} F ⟶ R̂[t, t^{−1}]ϕ̂ ⊗_{R} F u ⊗ x ↦ ut ⊗ f(x).

This is well-defined because of the following calculation

ut ⊗ f(rx) = ut ⊗ ϕ(r)f(x) = utϕ̂(i(r)) ⊗ f(x) = ui(r)t ⊗ f(x).

Now define the upper horizontal map by

τ ([f : F ⟶ ϕ^{∗}F]) := [r_{t} ⊗ f : R̂[t, t^{−1}]ϕ̂ ⊗_{R} F ⟶ R̂[t, t^{−1}]ϕ̂ ⊗_{R} F ] .  (7.5)

One easily checks that this is compatible with the relations appearing in Definition 1.2. The m-th component τ_{m} appearing in the lower horizontal map is given on representatives by

τ_{m}([r]) := [t^{m}i(r)].  (7.6)

One easily checks that the diagram 7.1 commutes.

Remark 7.7. The construction of the ring R̂[t, t^{−1}]ϕ̂ comes from the mapping torus construction as explained next. Let f : X ⟶ X be an endomorphisms of a connected finite CW -complex X. The mapping torus T_{f} is obtained from the cylinder X × [0, 1] by identifying the bottom and the top by f, i.e. by the identification (y, 1) ∼ (f(y), 0) for y ∈ X. If we put R = π_{1}(X) and ϕ : R → R to be the ring homomorphism induced by f, then R̂[t, t^{−1}]ϕ̂ is just ℤπ_{1}(T_{f}).

Remark 7.8. An interesting problem is to detect the kernels of the maps appearing in the diagrams 7.1. The point for the map dim ×η for dim as defined in 1.5 is that u(f) ∈ U(f) defined in 2.2 is the invariant we want to know and that its image under η which is the collection of the generalized Lefschetz invariants λ(f^{n}) ∈ Λ(f^{n}) for n ≥ 1 of Example 3.9 is easier to compute and the vanishing of λ(f^{n}) has a clear interpretation in terms of fixed points of f^{n} as explained in Example 3.5. If dim ×η would be injective then one could use the generalized Lefschetz invariants λ(f^{n}) and the Euler characteristic to compute u(f). Notice that dim ×η is injective if R is commutative and π = id by Example 1.7 or if (R, ϕ) = (ℂ, c) by Theorem 1.9 but we have no idea what happens in general.

For the map τ resp. U(i) injectivity would mean that one does not loose information by the mapping torus approach. The advantage of the mapping torus approach is that for the target of the maps τ the endomorphism of the ring is the identity. Of course one has to pay the price that the ring R̂[t, t^{−1}]ϕ̂ is more complicated than R. At least we can show injectivity of the lower map τ in Lemma 7.9 below.

Lemma 7.9. The lower horizontal map τ (see 7.6) in diagram 7.1

τ : ∏_{m≥1} Λ(R, ϕ^{m}) ⟶ ∏_{m≥1} Λ(R̂[t, t^{−1}]ϕ̂, id)

is injective if Λ(R, ϕ^{m}) contains no m-torsion for m ≥ 1.

Proof. We have to show for m ≥ 1 that the map

τ_{m} : Λ(R, ϕ^{m}) ⟶ Λ(R̂[t, t^{−1}]ϕ̂, id)  [r] ↦ [t^{m}i(r)]  (7.10)

is injective. Define

τ′_{m} : Λ(R, ϕ^{m}) ⟶ Λ(R̂[t, t^{−1}]ϕ̂^{m}, id)  [r] ↦ [ti(r)].  (7.11)

We obtain a ring homomorphism

j : R̂[t, t^{−1}]ϕ̂^{m} ⟶ R̂[t, t^{−1}]ϕ̂  ∑_{n} t^{n}a_{n} ↦ ∑_{n} t^{nm}a_{n}.

It induces a map

j_{∗} : Λ(R̂[t, t^{−1}]ϕ̂^{m}, id) ⟶ Λ(R̂[t, t^{−1}]ϕ̂, id).  (7.12)

One easily checks that τ_{m} is the composition j_{∗} ∘ τ′_{m}. It suffices to show for m ≥ 1 that τ′_{m} is injective and the kernel of j_{∗} is m-torsion.

We begin with injectivity of τ′_{m}. Obviously it suffices to do this for m = 1, otherwise substitute ϕ by ϕ^{m}. Suppose that [ti(r)] ∈ Λ(R̂[t, t^{−1}]ϕ̂, id) is trivial. Hence we can rewrite ti(r) in R̂[t, t^{−1}]ϕ̂ as a finite sum of elements of the form uv − vu for u, v ∈ R̂[t, t^{−1}]ϕ̂. Inspecting the coefficients of t^{1} one sees that i(r) is a finite sum of elements of the form ϕ̂^{a}(r̂)ŝ− ϕ̂^{1−a}(ŝ)r̂ for r̂, ŝ ∈ R̂ and a ∈ ℤ. Since for r̂, ŝ ∈ R̂ we have

(ϕ̂^{a}(r̂)ŝ− ϕ̂^{1−a}(ŝ)r̂) − (ϕ̂^{a−1}(r̂)ϕ̂^{−1}(ŝ) − ϕ̂^{1−(a−1)}(ϕ̂^{−1}(ŝ))r̂)
= ϕ̂^{a}(r̂)ŝ− ϕ̂^{a−1}(r̂)ϕ̂^{−1}(ŝ)
= (ϕ̂^{a}(r̂)ŝ− ŝϕ̂^{a−1}(r̂)) + (ŝϕ̂^{a−1}(r̂) − ϕ̂^{a−1}(r̂)ϕ̂^{−1}(ŝ)) ,

we can show by induction over a that the element i(r) is a finite union of elements of the shape ϕ̂(r̂)ŝ − ŝr̂ for r̂, ŝ ∈ R̂. For any element r̂ ∈ R̂ there is r ∈ R and a non-negative integer k with ϕ̂^{k}(r̂) = i(r). Hence there is a non-negative integer l such that i(ϕ^{l}(r)) is a finite union of elements of the shape i (ϕ(r)s − sr) for r, s ∈ R. Notice that the kernel of i : R ⟶ R̂ is the union ∪_{n≥1} ker(ϕ^{n}) and that ϕ^{n}(r) − r for n ≥ 0 represents zero in Λ(R, ϕ) because of

ϕ^{n}(r) − r = ϕ^{n}(r)1 − 1ϕ^{n−1}(r) + … + ϕ(r)1 − 1r.

Hence the map τ′_{m} of 7.11 is injective.

To show that the kernel of the map j_{∗} defined in 7.12 is m-torsion, we define a map

j^{∗} : Λ(R̂[t, t^{−1}]ϕ̂, id) ⟶ Λ(R̂[t, t^{−1}]ϕ̂^{m}, id)
[u] ↦ tr_{R̂[t,t^{−1}]ϕ̂^{m},id} (j^{∗}r_{u} : j^{∗}R̂[t, t^{−1}]ϕ̂ ⟶ R̂[t, t^{−1}]ϕ̂) ,

where for u ∈ R̂[t, t^{−1}]ϕ̂ the R̂[t, t^{−1}]ϕ̂-map r_{u} is given by right multiplication with u, j^{∗} denotes restriction with the ring homomorphism j and tr_{R̂[t,t^{−1}]ϕ̂^{m},id} has been introduced in

7.3. Since r_{uv−vu} = r_{v} ∘ r_{u} − r_{u} ∘ r_{v} holds and j^{∗}R̂[t, t^{−1}]ϕ̂ is the free R̂[t, t^{−1}]ϕ̂^{m}-module with basis t^{0}, t^{1}, …, t^{m−1}, the map j^{∗} is well-defined. One easily checks that for u ∈ R̂[t, t^{−1}]ϕ̂^{m} j^{∗}r_{j(u)} is conjugated to the direct sum of m-copies of the R̂[t, t^{−1}]ϕ̂^{m}-map r_{u} : R̂[t, t^{−1}]ϕ̂^{m} ⟶ R̂[t, t^{−1}]ϕ̂^{m} given by right multiplication with u. Since tr_{R̂[t,t^{−1}]ϕ̂^{m},id}(r_{u}) = u, we conclude that j^{∗} ∘ j_{∗} = m · id. Hence the kernel of j_{∗} is m-torsion. This finishes the proof of Lemma 7.9.

Remark 7.13. Notice that there is an obvious m-fold covering p_{m} : T_{f^{m}} ⟶ T_{f} for an endomorphism f : X ⟶ X of a connected finite CW -complex. Let i_{m} : X ⟶ T_{f^{m}} and i : X ⟶ T_{f} be the canonical inclusions. We obtain morphisms i_{m} : (X, f^{m}) ⟶ (T_{f^{m}}, \overline{f^{m}}) and p_{m} : (T_{f^{m}}, \overline{f^{m}}) ⟶ (T_{f}, \overline{f}^{m}) in End(C). The composition p_{m} ∘ i_{m} is just the morphism i : (X, f) ⟶ (T_{f}, \overline{f}^{m}). In particular we obtain a factorization of Λ(i) into Λ(p_{m}) ∘ Λ(i_{m}). One easily checks that this corresponds to the factorization of τ_{m} into j_{∗} ∘ τ′_{m} appearing in the proof of Lemma 7.9. Moreover, the map j^{∗} is just the transfer map associated to the fibration p_{m} : T_{f^{m}} ⟶ T_{f} defined in 5.2.

The assumption in Lemma 7.9 that Λ(R, ϕ^{m}) contains no m-torsion is always satisfied in the case where R is an integral group ring ℤπ and ϕ given by an endomorphism of π because then Λ(ℤπ, ϕ^{m}) is the free abelian group generated by the ϕ^{m}-conjugacy classes.

Remark 7.14. We have introduced several homomorphisms with source U(f) which can be used to detect elements in U(f). Notice that the homomorphism 3.2 and the homomorphism 3.11 factorize over η. The map 3.15 factorizes over η ∘ τ = τ ∘ η. However, it seems to be very unlikely that the map ρ of 3.13 factorizes over η or τ.

References

[1] Almkvist, G.: “The Grothendieck ring of the category of endomorphisms”, J. of Algebra 28, 375 - 388 (1974)

[2] Brown, R.F.: “The Lefschetz fixed point theorem”, Scott, Foresman and Company (1971)

[3] Burghelea, D., Friedlander, L. and Kappeler, T.: “Torsion for manifolds with boundary and glueing formulas”, preprint (1996)

[4] Burghelea, D., Friedlander, L., Kappeler, T. and McDonald, P.: “Analytic and Reidemeister torsion for representations in finite type Hilbert modules”, Geometric Analysis and Functional Analysis 6, 751 - 859 (1996)

[5] Carey, A.L. and Mathai, V.: “L^{2}-acyclicity and L^{2}-torsion invariants”, Contemporary Mathematics 105, 141–155 (1990)

[6] Cohen, M.M.: “A course in simple homotopy theory”, Graduate Texts in Mathematics 10, Springer (1973)

[7] Deseyve, M.: “Verallgemeinerte Lefschetz Zahlen”, Diplomarbeit, Mainz (1994)

[8] Dodziuk, J. and Mathai, V.: “Approximating L^{2}-invariants of amenable covering spaces: A combinatorial approach”, preprint (1996)

[9] Dold, A.: “The fixed point index of fibre-preserving maps”, Inventiones Mathematicae 25, 281–297 (1974)

[10] Dold, A.: “The fixed point transfer of fibre preserving maps”, Math. Z. 148, 215–244 (1976)

[11] Felśhtyn, A.,L. and Hill, R.: “Dynamical zeta functions, Nielsen theory and Reidemeister torsion”, in Proc. “Nielsen theory and dynamical systems”, Mt. Holyohe College 1992, editor: Mc Cord, C.K., Contemproray Mathematics 152, 141 - 157 (1993)

[12] Felśhtyn, A.,L. and Hill, R.: “The Reidemeister Zeta function with applications to Nielsen theory and connections to Reidemeister torsion”, K-theory 8, 367 - 393 (1994)

[13] Fried, D.: “Growth rate of surface homeomorphisms and flow equivalence”, Ergod. Th. and Dynam. Syst. 5, 539 - 563 (1985)

[14] Geoghegan, R. and Nicas, A.: “Lefschetz trace formula, zeta functions and torsion in dynamics”, in Proc. “Nielsen theory and dynamical systems”, Mt. Holyohe College 1992, editor: Mc Cord, C.K., Contemporay Mathematics 152, 141 - 157 (1993)

[15] Geoghegan, R. and Nicas, A.: “Parametrized Lefschetz-Nielsen fixed point theory and Hochschild homology traces”, in Proc. “Nielsen theory and dynamical systems”, American J. of Mathematics 116, 397 - 446(1994)

[16] Grayson, D.: “The K-theory of endomorphisms”, J, of Algebra 48, 439 - 446 (1977)

[17] Jiang, B.: “Estimation of the Nielsen numbers”, Chines Math. 5, 330 - 339 (1964)

[18] Jiang, B.: “Lectures on Nielsen fixed point theory”, Contemp. Math. 14, AMS (1983)

[19] Jiang, B.: “Estimation of the number of periodic orbits”, Topologie und nichtkommutative Geometrie 65, Mai 1993, Heidelberg (1993)

[20] Jiang, B. and Wang, S.: “Lefschetz numbers and Nielsen numbers for homeomorphisms and aspherical manifolds”, Topology Hawai 1990, editor: Dovermann, 119-136 (1992)

[21] Jiang, B. and Wang, S.: “Twisted topological invariants associated with representations”, in “Topics in Knot theory”, editor: Bozhüyük, 211 - 227 (1993)

[22] Laitinen, E. and Lück, W.: “Equivariant Lefschetz classes”, Osaka J. Math. 26, 491 - 525 (1989)

[23] Lott, J.: “Heat kernels on covering spaces and topological invariants”, J. of Diff. Geom. 35, 471 - 510 (1992)

[24] Lott, J. and Lück, W.: “L^{2}-topological invariants of 3-manifolds”, Inventiones Math. 120, 15 -60 (1995)

[25] Lück, W.: “The geometric finiteness obstruction”, Proc of the LMS 54, 367 - 384 (1987)

[26] Lück, W.: “The transfer maps induced in the algebraic K_{0}- and K_{1}-groups by a fibration I”, Math. Scand. 59, 93 - 121 (1986)

[27] Lück, W.: “The transfer maps induced in the algebraic K_{0}- and K_{1}-groups by a fibration II”, J. of Pure and Applied Algebra 45, 143 - 169 (1987)

[28] Lück, W.: “Transformation groups and algebraic K-theory”, Lecture Notes in Mathematics vol. 1408 (1989)

[29] Lück, W.: “L^{2}-torsion and 3-manifolds”, Conference Proceedings and Lecture Notes in Geometry and Topology Volume III “Low-dimensional topology”, Knoxville 1992, editor: Klaus Johannson, International Press, 75- 107 (1994)

[30] Lück, W.: “L^{2}-Betti numbers of mapping tori and groups”, Topology 33, 203 - 214 (1994)

[31] Lück, W.: “L^{2}-invariants of regular coverings of compact manifolds and CW - complexes”, to appear in handbook of geometry“, editors: Davermann, R.J. and Sher, R.B., Elsevier (1998)

[32] Lück, W. and Ranicki, A.: “The surgery transfer”, Konferenzbericht der Göttinger Topologie Tagung 1987 (editor : tom Dieck), Lecture Notes in Mathematics 1361, 167 - 246, Springer (1988)

[33] Lück, W. and Ranicki, A.: “Surgery obstructions of fibre bundles”, J. of Pure and Applied Algebra 81, 139 -189 (1992)

[34] Lück, W. and Rothenberg, M.: “Reidemeister torsion and the K-theory of von Neumann algebras”, K-theory 5, 213–264 (1991)

[35] Lück, W. and Schick, T.: “L^{2}-torsion of hyperbolic manifolds of finite volume”, preprint, Münster (1997)

[36] Lydakis, M.G.: “Fixed point problems, equivariant stable homotopy theory, and a trace map for the algebraic K-theory of a point”, Topology 34, 959–999 (1995)

[37] Mathai, V.: “L^{2}-analytic torsion”, J. of Funct. Analysis 107, 369 - 386 (1992)

[38] Milnor, J.: “Infinite cyclic coverings”, Proc. Conf on the Topology of Manifolds, 115 - 133 (1968)

[39] Okonek, C.: “Bemerkungen zur K-Theorie äquivarianter Endomorphismen” Arch. Math. 40, 132 - 138 (1983)

[40] Pedersen, E.K. and Taylor, L.: “The Wall finiteness obstruction for a fibration”, Amer. J. Math. 100, 887 - 896 (1978)

[41] Reidemeister, K.: “Automorphismen von Homotopiekettenringen”, Math. Annalen 112, 586 ; 593 (1938)

[42] Wecken, F.: “Fixpunktklassen II”, Math. Annalen 118, 216 - 243 (1942)

Current address Wolfgang Lück Fachbereich Mathematik und Informatik Westfälische Wilhelms-Universität Münster Einsteinstr. 62 48149 Münster Bundesrepublik Deutschland email: lueck@math.uni-muenster.de FAX: 0251 8338370 internet: http://wwwmath.uni-muenster.de/math/u/lueck/

Version of September 18, 1998