Vol. 89, No. 2, 1980

Recent Issues
Vol. 330: 1
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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