The notion of degree for non-orientable manifolds mapping
by
Evgenij G. Skljarenko
Moscow State University, Moscow, Russia

Compact cohomology groups of manifolds in the higher dimension with coefficients in the orientation sheaves are used for a definition of the degree of manifolds proper mappings.

The degree deg f can be defined only for orientable mappings f, i.e. for mappings which are coordinated with corresponding orientation sheaves mappings. This deg f coincides with Hopf's Absolutgrad A(f). For a non-orientable mapping f of a manifold M there exists a single double covering p of M such that the composition fp is an orientable mapping. In this case the invariant A(f) is equal to 1/2 deg f.

The original definitions of A(f) are essentially complicated (see, for example, D. B. A. Epstein, Proc. London Math. Soc., (3), 16 (1966), 369-383). It was shown by Hopf however, that A(f) coincides with the minimal integer d for which there exist a properly homotopic to f mapping g and a ball D in the image of f, the inverse image of which by means of g is the union of d distinct balls in M with the restrictions of g on each of them being homeomorphisms.

The described connection between A(f) and the cohomological degree simplifies the investigation of A(f). Thus, it can be shown that A(gf) = A(g) A(f) for any composition of proper mappings f, g. For mappings of manifolds with connected boundaries A(f) = A(f'), where f' means the restriction of f on boundaries. Finally A(f) = A(g) for properly bordant mappings f, g, and A(f) = 0 if a mapping f is bordant to a constant mapping.

Date received: May 25, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caje-08.