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
f−1(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 : R2m−1→ 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.