A homology spectral sequence for submersions

Abstract

By a submersion we shall understand a C sur jection /: X-+Y between paracompact C manifolds with dim X > dim F, subject to the condition that the differential of / have maximal rank at all points. This implies that the fiber f y over any point yeY 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 f y 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 E 2 terms now come to be identified with certain groups H s , t (f; G) representing a homology functor from the category of submersions to the category of bigraded groups, which reduce of course to H S (Y; H t (F; G)) in the classical case.