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Coincidence index for fiber-preserving maps: An approach to stable cohomotopy. (English) Zbl 0551.55002
The purpose of this paper is to define, in the same spirit as 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).

MSC:
 55M20 Fixed points and coincidences in algebraic topology 55Q10 Stable homotopy groups 55M25 Degree, winding number 55Q55 Cohomotopy groups 55P91 Equivariant homotopy theory in algebraic topology
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References:
 [1] Dold, A. ?Lectures on Algebraic Topology? Berlin-Heidelberg-New York: Springer 1972 · Zbl 0234.55001 [2] Dold, A. The Fixed Point Index of Fibre-Preserving Maps Inventiones math. 25, 281-297 (1974) · Zbl 0284.55007 [3] Dold, A. The Fixed Point Transfer of Fibre-Preserving Maps Math. Z. 148, 215-244 (1975) · Zbl 0329.55007 [4] Sch?fer, B. ?Fixpunkttransfer f?r stetige Familien von ANR-Ra?men; Existenz and axiomatische Charakterisierung;? Dissertation, Heidelberg, 1981
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