Session 35. Topological fixed point theory and related topics

Codimension one coincidence Indices for spin \(PL\) manifolds

Donco Dimovski, University "Ss Cyril and Methodius", Skopje, Republic of Macedonia
Using the results and techniques about one-parameter fixed point theory from [3], one-parameter fixed point indices from [1], and the geometric description of spin manifolds and spin structures based on [2], two indices for codimension one coincidences are defined, as follows. Let \(F, \, G: X \to Y\) be \(PL\) maps where \(X\) and \(Y\) are and spin, closed, connected \(PL\) manifolds, \(X\) is \((n + 1)\)-dimensional and \(Y\) is an \(n\)-dimensional, \(n \geq5\). A coincidence of \(F\) and \(G\) is a point a \(X\) such that \(F(a)=G(a)\). The set of all the coincidences is denoted by Coin(F,G). For a family \(V \) of isolated circles of coincidences of \(F\) and \(G\), we define two indices: \(ind_1(F,G; V)\) - which is an element in the first homology group \(H_1(E)\), where \(E\) is the space of paths in \(X \times Y\) from the graph of \(F\) to the graph of \(G\); and \(ind_2(F,G; V\)) - which is an element in the group \(\textbf{Z}_2\) with two elements. We prove that for a family \(V\) of isolated circles of coincidences of \(F\) and \(G\) in the same coincidence class there is a neighborhood \(N\) of \(V\) and a homotopy from \(F\) to \(H\) rel \(X\backslash N\) such that \(Coin(H,G)=Coin(F,G)\backslash V\) if and only if \(ind_1(F,G;V) = 0\) and \(ind_2(F,G; V) = 0\).
References
  1. D. Dimovski, One-parameter fixed point indices , Pacif. J. of Math., Vol. 161. No. 2, 1994, 263-297.
  2. D. Dimovski, Canonical Embeddings of \(\,S^1\times\Delta^{n-1} \) into orientable \(n\)-dimensional closed \(PL\) manifolds for \(n>4\), Top. And its Applic., Volume 160, Issue 17, 2013, 2141-216.
  3. D. Dimovski, R. Geoghegan, One-parameter Fixed Point Theory , Forum Math. 2, 1990, 125-154.
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