Coincidence theory for maps from a complex into a manifold
type: Item
instance of: scholarly article
title: Coincidence theory for maps from a complex into a manifold
zbMATH Open document ID: 0927.55003
DOI: 10.1016/S0166-8641(97)00231-9
published in: Topology and its Applications
publication date: 1999-12-13T00:00:00Z
review text: For maps \(f,g:X\to Y\), the author denotes by \(\mu(f,g)\) the minimum number of coincidences of \(f'\) and \(g'\), that is, solutions to \(f'(x)= g'(x)\), among all \(f'\) homotopic to \(f\) and \(g'\) homotopic to \(g\). By analogy with fixed point theory, \(\mu(f,g)\) would be studied by partitioning the coincidence set of \(f\) and \(g\) into equivalence classes called coincidence classes and then a coincidence index would distinguish the essential coincidence classes as those of non-zero index and the Nielsen coincidence number \(N(f,g)\) defined as the number of essential coincidence classes. If \(X\) and \(Y\) are closed orientable \(n\)-manifolds, then an integer coincidence index is readily defined for any isolated set of coincidences [see \textit{J. W. Vick}, Homology theory, Grad. Texts Math. 145 (1994; Zbl 0789.55004), p. 177] so that \(N(f,g) \leq\mu (f,g)\) with equality when \(n\geq 3\) [\textit{H. Schirmer}, J. Reine Angew. Math. 194, 21-39 (1955; Zbl 0066.41701)]. Without assuming orientability of the \(n\)-manifolds, it is possible to define an index for coincidence classes (but not for arbitrary isolated coincidence sets) which takes values either in the integers or \(Z_2\), depending in the coincidence class, so that the same result holds [\textit{R. Dobrenko} and \textit{J. Jezierski}, Rocky Mt. J. Math. 23, No. 1, 67-85 (1993; Zbl 0787.55003)]. In the present paper, the author initiates the study of a suitable Nielsen theory in the much more general setting where \(Y\) is an \(n\)-manifold and \(X\) is assumed only to be a finite \(n\)-dimensional polyhedron. Following the obstruction theoretic approach used for the fixed point index used by \textit{E. Fadell} and \textit{S. Husseini} in [Topology 20, 53-93 (1981; Zbl 0453.55002)], the author defines an index of a coincidence class that takes its value in a group that depends on the coincidence class. The Nielsen number \(N(f,g)\) is then defined to be the number of coincidence classes with no…
reviewed by: Robert F. Brown
Mathematics Subject Classification ID: 55M20
zbMATH DE Number: 1307882
zbMATH Keywords: obstruction class
zbMATH Keywords: coincidence index
zbMATH Keywords: coincidence class
zbMATH Keywords: Nielsen number
author: Daciberg Lima Gonçalves
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1307882