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 7TI(B) on the homology of the fibre. This was noticed by D. R. Anderson [1], who produced formulae for the finiteness obstruction of E in case the fibration is a flat bundle [1, 2]. 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(ZT1 (B )). In case 7T1(E)-T1(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 7T1(X) on S*(X) making it a free Z7rTX module. Given a ring A and a ring homomorphism Z7T1X-4A, we say that X is A-dominated iff