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On the coincidence points of mappings of the torus into a surface. (English. Russian original) Zbl 1108.55002

Geometric topology and set theory. Collected papers. Dedicated to the 100th birthday of Professor Lyudmila Vsevolodovna Keldysh. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica. Proceedings of the Steklov Institute of Mathematics 247, 9-27 (2004); translation from Tr. Mat. Inst. Steklova 247, 15-34 (2004).
Let \(f, g\colon M\to N\) be two maps between manifolds \(M\) and \(N\). The coincidence Nielsen number \(N(f,g)\), which is the number of essential coincidence classes of \(f\) and \(g\), serves as a lower bound for the number of coincidence points of all map pairs in the homotopy class of the map pair \((f,g)\), i.e. \(N(f,g)\leq \min_{f'\simeq f, g'\simeq g} \sharp\{x | f'(x) =g'(x)\}\). A natural question is whether the two maps \(f\) and \(g\) have the Wecken property. This means that the above inequality turns out to be an equality. Usually, the dimensions of the domain \(M\) and the target \(N\) are assumed to be the same. In this situation, the Wecken property holds for all maps if \(\dim M =\dim N\neq 2\). The exceptional dimension \(2\) shows the typical difficulty in low dimensional topology. The Wecken property was shown in some special cases. The paper under review gives a positive answer for the Wecken property if the domain manifold \(M\) is a torus. Moreover, the authors show that each essential coincidence class has index \(\pm 1\) if \(N\) is an orientable surface \(\neq S^2\) and has semi index \(1\) if \(N\) is a non-orientable surface \(\neq \mathbb{R} P^2\).
For the entire collection see [Zbl 1087.55002].

MSC:

55M20 Fixed points and coincidences in algebraic topology
57M60 Group actions on manifolds and cell complexes in low dimensions
57S25 Groups acting on specific manifolds
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