On the homology of fiber spaces

Abstract

Let (F, T, X, 7r) be a fiber space, with fiber F, base X, total space T, and fiber map 7r. A general problem of great interest is that of computing the homology groups of one of the spaces involved, usually F or T, in terms of the homology groups of the other two spaces and, perhaps, some other invariants of the fiber In this paper we show how the Lusternik-Schnirelmann of X enters into this problem and affects the relations which may exist between the homology groups of F, T, and X. Our main results are stated as theorems and corollaries in ??3 and 4 of this paper, and are summarized here. Let OX denote the space of loops on X. If cat(X) S k, we obtain a spectral sequence, Ar, which relates H(F) and H(QX) with H(T) and for which the differentials, dr, and groups, 4, vanish if r, p?k. If cat(X) !2, we obtain an infinite exact sequence relating H(X), H(F), and H(T) which generalizes the Wang sequence. This allows us to compute the additive structure of H(QX) and to partially determine the Pontryagin ring H*(QX). We also consider the Leray-Serre spectral sequence of the fiber space and essentially compute all the differentials if cat(X) ? 2. Our method is to replace the chain group of T by a twisted tensor product, WAA QC(F), where C( Y) denotes the group of chains of Y, A = C(QX), and WA is the bar construction on A. We then apply certain results of [5] which relate cat(X) and WA. The necessary definitions and preliminary material are covered in ??1 and 2, while the proofs of the main theorems are in ?5. Some related results are contained in [6]. In that paper we also obtain some results involving the category of a map, similar to those obtained here by using the category of a space.