Vol. 89, No. 2, 1980

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A homology spectral sequence for submersions

Patrick C. Endicott and J. Wolfgang Smith

Vol. 89 (1980), No. 2, 279–299

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

E2s,t ≈ Hs(Y;Ht(F ;G ))

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.

Mathematical Subject Classification 2000
Primary: 55T10
Secondary: 57R35
Received: 23 August 1978
Published: 1 August 1980
Patrick C. Endicott
J. Wolfgang Smith