On the Wall finiteness obstruction for the total space of certain fibrations

The problem of computing the Wall finiteness obstruction for the total space of a fibration p: E -* B in terms of that for the base and homological data of the fiber has been considered by D. R. Anderson and by E. K. Pedersen and L. R. Taylor. We generalize their results and show how the problem is related to the algebraically defined transfer map q*: K0(Z7r,(B)) -KO(Z-r,(E)), q = p*: vl(E) ,rl(B), whenever the latter is defined. 0. Introduction. Let p: E -* B be a Serre fibration with fiber F. Assume that B is finitely dominated and that F has the homotopy type of a finite complex. Let w(B) E Ko(Z T"(B)) be Wall's finiteness obstruction for B [19]. Since E is also finitely dominated [10], one also has wT(E) E K0(ZrTl(E)). We study the relationship between w(E) and w(B). Assume given a factorization of p* i I((E ) ST --> T(B) with s onto, and Ker(qp) = v of type (FP) (i.e. Z, viewed as a Zv module with trivial v action, admits a finite resolution by finitely generated, projective Zv modules). We compute s*(wi(E)) under the assumption that a certain covering F of a component of F has finitely generated integral homology. The description involves the transfer map induced by (p and the integral representations Hi(F; Z) of the group ?T, see Theorem 4.1 for details. Taking v trivial, i.e. ?T = Im(p*: T1(E) (B)), one recovers the main result of Pedersen and Taylor [13]. Theorem 4.1 has e.g. the following. COROLLARY A. Let v be a group for which BP is a finite cowmlex. If p: E B has fiber F = BP and 7r1(F) -?1(E) is injective then w-(E) = p*(w-(B)) where qp*: KO(Z?T1(B)) -* KO(Z'Ir1(E)) is the transfer map induced by (p = p*: 7r1(E) Tr1(B). Received by the editors November 29, 1978 and, in revised form, May 15, 1979. AMS (MOS) subject classifications (1970). Primary 55F05, 57C05, 57C10; Secondary 18F25, 18G05, 18G10.