The mod \(H\) Nielsen theory and the \(q\)-Nielsen theory
Let \(X\) be a compact connected polyhedron and \(x_0\in X\) and \(H\) be a normal subgroup of \(\pi_1(X,x_0)\) and \(p:\tilde{X}\to X\) the universal covering. Two fixed points \(x_1,x_2\) of \(f:X\to X\) are said to be in the same mod \(H\) fixed point class if there exists a path \(c\) from \(x_1\) to \(x_2\) such that \([b*c*(f*c)^{-1}*b^{-1}]\in H\) where \(b\) is a path from \(x_0\) to \(x_1\). The author proves that \(x_1\) and \(x_2\) are in the same mod \(H\) fixed point class iff there exists a path \(c\) from \(x_1\) to \(x_2\) such that \([c*(f\circ c)^{-1}]\in H\). Following \textit{M. H. Woo} and \textit{J.-R. Kim} [J. Korean Math. Soc. 29, No. 1, 117--125 (1992; Zbl 0764.55006)] for a map \(q:X\to Y\) two fixed points \(x_1,x_2\) of \(f\) are said to be \(q\)-equivalent if there exists a path \(c\) from \(x_1\) to \(x_2\) such that \(q\circ (f\circ c)\) and \(q\circ f\) are homotopic via a homotopy keeping the end points fixed. Fix a lift \(\tilde{f}\) to \(\tilde{X}\) of \(X\) such that \(x_0\in p(\text{Fix}(\tilde{f}))\). Two lifts \(\tilde{f}'= \alpha\circ\tilde{f}\) and \(\tilde{f}''=\beta\circ\tilde{f}\) with \(\alpha,\beta\in\pi_1(X,x_0)\) are said to be \(q\)-equivalent if there exists a \(\gamma\in\pi_1(X,x_0)\) such that \(q_{\#}(\beta)=q_{\#}(\gamma)\circ q_{\#}(\alpha)\circ q_{\#}(\tilde{f}_{\#}(\gamma^{-1}))\). The corresponding equivalence is denoted by \([\tilde{f}']_q\). Define the fixed point class \(\mathbb{F}_q:=\bigcup_{\tilde{f}''\in[\tilde{f}']_q}p(\text{Fix}(\tilde{f}''))\). The author proves that two fixed points \(x_1\) and \(x_2\) belong to the same class \(\mathbb{F}_q\) iff \(x_1\) and \(x_2\) are \(q\)-equivalent.
scientific article; zbMATH DE number 6162217
scientific article