Coincidence index for fiber-preserving maps: An approach to stable cohomotopy
The purpose of this paper is to define, in the same spirit as \textit{A. Dold}'s fixed point index for fiber-preserving maps [Invent. Math. 25, 281-297 (1974; Zbl 0284.55007)] a coincidence index for certain pairs of maps between Euclidean neighborhood retracts over a metric space B in any cohomology theory. An adequate geometric equivalence relation between two such coincidence situations is used to define elements of groups \(FIX^ k(B)\) and \(FIX^ k(B,A)\) for k an integer and A closed in B. It is shown that these groups constitute a generalized multiplicative cohomology theory and that the index determines a natural transformation of cohomology theories, which in the case of stable cohomotopy turns out to be an isomorphism. This approach to stable cohomotopy can be generalized to the equivariant and to the parametrized case (over B).
coincidence index for pairs of maps between Euclidean neighborhood retracts over metric space
generalized multiplicative cohomology theory
scientific article; zbMATH DE number 3877965