Fiber homology and orientability of maps
J. Wolfgang Smith · Pacific Journal of Mathematics · 1980
In this paper we introduce a concept of fiber homology for an arbitrary map f\X-»Y and coefficient module G.This is a graded module denoted by H*(f*; (?) which reduces to H*(F; G) when / represents an orientable fiber bundel with standard fiber F. The concept of fiber homology permits us also to define a generalized notion of orientability, and these ideas turn out to be useful in the study of submersions.Our main theorem (obtained by means of a spectral sequence) asserts that if the fibers of a submersion f:X-*Y are acyclic in dimensions smaller than q, then the rank r q of the fiber homology H q (f%; G) is bounded above by the sum of the q and (g+l)-dimensional Betti numbers of X and Y, respectively.In the orientable case, the g-dimensional Betti number of an arbitrary fiber f"Ήy) is bounded above by r q9 and therefore also by the aforementioned sum.This leads to a number of more specialized results.For example, it is shown that the fibers of an orientable submersion /: R 2m ~1->S m must be either acyclic or homology spheres, and moreover, the subspace of points in S m corresponding to the spherical fibers must have the homology of a point.1* Basic concepts.Let f:X->Y denote a continuous map between topological spaces.By a tubular neighborhood belonging to / we will understand a homeomorphism Φ: B x F & V where B is an open connected subspace of Y, F a compact space and V a subset of X, such that foφ is the projection B x F-* B. Given a point y e B, we will write *;= vnf y where f y denotes the preimage of y under /, and given two points y, y' e B, we let Φfl Fy P* Fy, denote the homeomorphism induced by Φ.The diagram