Bounds for fixed points on Seifert manifolds (Q452086)

\textit{B. J. Jiang, S. D. Wang} and the author [Algebr. Geom. Topol. 11, No. 4, 2297--2318 (2011; Zbl 1232.55006)] defined the ``characteristic'' \(\text{chr}(f,\mathbf{F})\) for a fixed point class \(\mathbf{F}\) of a map \(f:M\to M\) as \(\text{chr}(f,\mathbf{F}):=1-\text{rank}(f,\mathbf{F})\) where \(\text{rank}(f,\mathbf{F})\) denotes the rank of the fixed subgroup of \(f_\#\) in \(\pi_1(M,x)\) for any \(x\in\mathbf{F}\) with the exception that \(\text{chr}(f,\mathbf{F})=\chi(S)\) if the fixed subgroup is of the form \(\pi_1(S)\) for some closed hyperbolic surface \(S\subset M\). Here, the author deals with the case where \(f\) is a homeomorphism and \(M\) is a compact connected orientable Seifert manifold (with or without boundary) with hyperbolic orbifold \(X(M)\). Then it is proved that \(\text{ind}(\mathbf{F})\leq\text{chr}(\mathbf{F})\) for every essential fixed point class \(\mathbf{F}\) of \(f\). Write \(s(\mathbf{F}):=\text{ind}(\mathbf{F})+\text{chr}(\mathbf{F})\) for an essential fixed point class of \(f\). Then we have that \(\sum_{s(\mathbf{F})<0}s(\mathbf{F})\geq B\) where \(B:=4(3-\text{rank}(\pi_1(M))\) if \(M=F\times S^1\) for a closed surface \(F\) and \(B=4(2-\text{rank}(\pi_1(M))\) else.