Coincidence index for fiber-preserving maps: An approach to stable cohomotopy
type: Item
instance of: scholarly article
title: Coincidence index for fiber-preserving maps: An approach to stable cohomotopy
zbMATH Open document ID: 0551.55002
DOI: 10.1007/BF01174595
author: Carlos Prieto
published in: Manuscripta Mathematica
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155013
review text: 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).
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 55Q10
Mathematics Subject Classification ID: 55M25
Mathematics Subject Classification ID: 55Q55
Mathematics Subject Classification ID: 55P91
zbMATH DE Number: 3877965
zbMATH Keywords: coincidence index for pairs of maps between Euclidean neighborhood retracts over metric space
zbMATH Keywords: generalized multiplicative cohomology theory
zbMATH Keywords: stable cohomotopy
MaRDI profile type: Publication
cites work: Q5643279
cites work: The fixed point index of fibre-preserving maps
cites work: The fixed point transfer of fiber-preserving maps
OpenAlex ID: W2015198017
description: scientific article; zbMATH DE number 3877965