By a submersion we shall
understand a C∞ surjection f : X → Y between paracompact C∞ manifolds with
dimX ≥dimY , subject to the condition that the differential of f have maximal rank
at all points. This implies that the fiber fy over any point y ∈ Y will be a
smooth regularly imbedded submanifold of X. Differentiable fiber bundles
constitute a special class of submersions, characterized by the existence of local
product structures, and in this particular case all fibers fy are homeomorphic
to a standerd fiber F. The central result in the homology theory of fiber
bundles asserts the existence of a convergent spectral sequence whose E∞ term
is the bigraded group associated to some filtration of H∗(X;G)1, and for
which
in case the bundle is orientable over G. In the present paper this result is generalized
to arbitrary submersions. The E2 terms now come to be identified with certain
groups Hs,t(f;G) representing a homology functor from the category of submersions
to the category of bigraded groups, which reduce of course to Hs(Y ;Ht(F;G)) in the
classical case.
