Minimum numbers and Wecken theorems in topological coincidence theory. I
Given a self-map \(f: X\to X\) on a compact polyhedron \(X\), the Nielsen number \(N(f)\) is a lower bound for the number of fixed point of all self-maps homotopic to \(f\). In the 1940's, Wecken proved that \(N(f)\) is sharp (i.e., there is a self-map which is homotopic to \(f\) and has \(N(f)\) fixed points) if \(X\) is a smooth manifold of dimension greater than \(3\). This result is called Wecken theorem. This paper deals with a more general setting: coincidence points of two maps \(f_1, f_2: M\to N\) between two manifolds without boundaries. The author provides several invariants which can be used to estimate from below the number of coincidence points or the number of components of the coincidence point sets, and obtains Wecken type theorems: under some assumptions these lowers bound are sharp. Although some of the invariants already appeared in previous works by the author, this paper contains a systematic treatment. All these invariants come from the path space \(E(f_1, f_2)\). The computation of the invariants is in general difficult, however the author in this paper gives many concrete and interesting examples where one can see some new approach to access these invariants. Some further theoretical applications are also provided. For instance, if \(M\) and \(N\) are spheres of special dimensions, respectively, the relations between the Kervaire invariant and the coincidence invariants are illustrated. These show the theoretical interaction among coincidence point theory and other mathematical fields.
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