The homology of twisted cartesian products

Abstract

Introduction. In a recent paper [2], E. H. Brown introduced the notion of a twisted tensor product. Briefly, the definition is as follows. Let K be a D.G.A. (differential, graded, augmented) coalgebra, A a D.G.A. algebra, and M a D.G.A. A-module. The twisted tensor product K 0 M of K with M is, except for the differential, the usual tensor product. The differential on K 0, 0 M is modified using a twisting cochain 4 in Hom(K, A). Now suppose p: E-*X is a fiber space (essentially of the Hurewicz type) with fiber F. Then C(X) is a D.G.A. coalgebra, C(QX) a D.G.A. algebra, and C(F) a D.G.A. C(QX)-module. Brown's main theorem states that there is a twisting cochain 4 in Hom(C(X), C(QX)) and a chain equivalence 41: C(X) X C(F) -> C(E). In another recent paper [1], Barratt, Gugenheim, and Moore define the twisted cartesian product of two simplicial sets (semi-simplicial complexes). If X and F are simplicial sets and G is a simplicial group acting on F, the twisted cartesian product of X and F, X XT F, coincides with the usual cartesian product except that the initial face is modified in terms of a twisting function r: X -G. It is proved in [1] that any simplicial fiber space p: EX with fiber F can, for the purposes of algebraic topology, be replaced by a twisted cartesian product X XT F. (The group G and the twisting function r: X >G are shown to exist.) Considering these two results, one might expect that an analogue of Brown's theorem could be proved for twisted cartesian products, explicitly defining the twisting cochain in terms of the twisting function(2). This is in fact done in Part I of this paper. In Part II, the explicit form of the twisting cochain is used to investigate fiber bundles over spheres. The homology and cohomology Wang sequences are derived and some partial results obtained describing the behavior of the maps in the cohomology Wang sequence with respect to cup products. I would like to express my gratitude to Professor Saunders MacLane for his patient assistance and encouragement during the preparation of this