Obstruction theory and minimal number of coincidences for maps from a complex into a manifold. (English) Zbl 1045.55001

Let \(f,g: X\to Y\) be maps and denote by \(MC[f, g]\) the minimum number of coincidences, that is solutions to \(f'(x)= g'(x)\), for all maps \(f'\) homotopic to \(f\) and \(g'\) homotopic to \(g\). In 1955, Schirmer proved that if \(X\) and \(Y\) are compact orientable manifolds of the same dimension \(n\geq3\), then the Nielsen coincidence number \(N(f, g)\), a natural generalization of the Nielsen fixed point number, calculates \(MC[f, g]\). Subsequent work of many researchers has sought to extend Schirmer’s result to more general classes of spaces. In this paper, \(Y\) is a compact but not necessarily orientable \(n\)-manifold and \(X\) is an \(n\)-dimensional finite simplicial complex. The authors demonstrate that the coincidence problem can be restricted to a subcomplex of \(X\) in which all maximal simplices are \(n\)-dimensional and there are no free \((n- 1)\)-faces. In this setting, a homotopy invariant \(NO(f,g;X)\) is defined that depends on the primary obstruction to deforming \(f\) and \(g\) to be coincidence free. If \(n\geq 3\) and \(X\) satisfies some mild geometric conditions, then \(NO(f,g;X)= MC[f,g]\). The authors use their theory to investigate \(W\{X, Y\}\), the maximum of \(MC[f, g] -N(f, g)\) over all homotopy classes of pairs of maps from \(X\) to \(Y\). They prove that, if \(Y\) is simply connected, then \(W\{X, Y\}\) is finite (though not in general zero). They show by example that \(W\{X,Y\}\) can be infinite. They also consider the interesting case where \(X\) is a finite union of closed \(n\)-manifolds.


55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)
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