Vol. 89, No. 2, 1980

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Fiber homology and orientability of maps

J. Wolfgang Smith

Vol. 89 (1980), No. 2, 453–470

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;G) which reduces to H(F;G) when f 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 rq of the fiber homology Hq(f;G) is bounded above by the sum of the q and (q + 1)-dimensional Betti numbers of X and Y , respectively. In the orientable case, the q-dimensional Betti number of an arbitrary fiber f1(y) is bounded above by rq, 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 f : R2m1 Sm must be either acyclic or homology spheres, and moreover, the subspace of points in Sm corresponding to the spherical fibers must have the homology of a point.

Mathematical Subject Classification 2000
Primary: 55R65
Received: 11 December 1978
Revised: 31 May 1979
Published: 1 August 1980
J. Wolfgang Smith