The Wall Finiteness Obstruction for a Fibration

Abstract

1. We wish to study the Wall finiteness obstruction for the total space of a fibration F → E → B. Such a study was first done by V. J. Lal [5], who neglected the action of π1(B) on the fibre. This was noticed by D. R. Anderson [2], who produced formulae for the finiteness obstruction of E in case the fibration is a flat bundle [2, 1]. Our main theorem gives a partial calculation of the general case: If B and E are finitely dominated and H∗(F ) is a finitely generated Z-module, we compute the image of the Wall obstruction of E in K0(Zπ1(B)). In case π1(E) → π1(B) is monic, we may refine this to compute the actual obstruction for E in terms of information on F and B. In this case the assumption that E is finitely dominated may be replaced by F being the homotopy type of a finite complex. We begin by establishing some terminology. Let X be a path connected space. We let S∗(X) denote the singular chain complex of the universal cover of X. There is an action of π1(X) on S∗(X) making it a free Zπ1X module. Given a ring Λ and a ring homomorphism Zπ1X → Λ, we say that X is Λ-dominated iff S∗(X)⊗Zπ1X Λ is chain homotopy equivalent to a complex P∗ of Λ-modules, where each Pi is a finitely generated projective Λ-module and all Pi are zero except for finitely many. If X is Λ-dominated, we can form σ(X; Λ) = ∑∞ i=−∞(−1)[Pi] in K0(Λ). Although many choices were necessary to form σ(X,Λ), it is easy to check that σ(X; Λ) is independent of all these choices. Also one sees that σ(X; Λ) is a homotopy invariant and natural with respect to homomorphisms Λ→ Λ1. We say that X is finitely dominated if X is Zπ1X-dominated (where Zπ1X → Zπ1X is the identity) and π1X is finitely presented. In this case we write σ(X;Zπ1X) = σ(X). Wall [7, 8] has proved that a CW complex is dominated by a finite CW complex iff it is finitely dominated and that it has the homotopy type of a finite CW complex iff σ(X) ∈ K0(Zπ1X) vanishes in K̃0(Zπ1X). We call the image of σ(X) in K̃0(Zπ1X) the Wall finiteness obstruction for X. We say that X is homologically finite if X is Z-dominated, where Zπ1X → Z is the natural map. It is easy to see that this is the case iff ∑∞ i=0Hi(X;Z) is a finitely generated abelian group. To describe the action of the fundamental group on the homology of the fibre we need a functor G(π), π a group, and a pairing G(π) × K0(Zπ1X) → K0(Zπ). We let G(π) be

Cite this paper

@inproceedings{Kjr2002TheWF, title={The Wall Finiteness Obstruction for a Fibration}, author={Erik Dahl Kj{\ae}r and L. Taylor and Vandana Lal}, year={2002} }