Coincidence theory for maps from a complex into a manifold
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 non-zero index in this sense. If \(X\) is an \(n\)-manifold, the groups are either the integers or \(Z_2\) and the author is able to recover the results both of Schirmer and of Dobrenko-Jezierski. He then shows that, by restricting \(f\) and \(g\) to subcomplexes, it is often sufficient to study \(\mu(f,g)\) in the case that \(X\) closely resembles an \(n\)-manifold, for instance \(X\) is homogeneous (all maximal simplices are \(n\)-dimensional) and \(n\)-dimensionally connected (each pair of \(n\)-simplexes can be joined by a chain of \(n\)-simplexes meeting in \((n-1)\)-dimensional faces). However, in the final section, the author presents several examples that indicate that the Nielsen number \(N(f,g)\) he has defined fails to estimate \(\mu(f,g)\) even when \(Y\) is simply connected and that the problem is sensitive to the geometry of \(X\) in the sense that changes of \(X\) even within homotopy type, indeed even when \(X\) is the homotopy type of a manifold, significantly affect the estimation problem. Thus, it appears that a concept more sensitive than \(N(f,g)\) will be needed if \(X\) is not a manifold. So, although the coincidence problem for maps from \(n\)-complexes to \(n\)-manifolds was identified at least as long ago as the 1981 Fadell-Husseini paper, it is still far from being solved, but the present paper makes a good start towards a solution.
scientific article; zbMATH DE number 1307882
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