Free degrees of homeomorphisms and periodic maps on closed surfaces
The free degree of a closed surface \(M\) is \(h(M)\) if, for every homeomorphism \(f:M\to M\), one of the iterates \(f\), \(f^ 2,\dots,f^{h(M)}\) has a fixed point and, moreover, there is a homeomorphism \(s:M\to M\) such that \(s\), \(s^ 2,\dots,s^{h(M)-1}\) are all fixed point free. The result of the paper is that if \(M=F_ g\), the orientable surface of genus \(g\), then \(h(F_ 2)=4\) and \(h(F_ g)=2g-2\) for \(g>2\) whereas if \(M=N_ g\), the nonorientable surface of genus \(g\), then \(h(N_ 3)=2\) and \(h(N_ g)=g-2\) for \(g>3\). Furthermore, the homeomorphism \(s\) (with \(s\), \(s^ 2,\dots,s^{h(M)-1}\) fixed point free) may be chosen to be periodic. For \(F_ g\), the construction of \(s\) makes use of cyclic branched covers and, for \(N_ g\), the construction of \(s\) depends on an explicit construction of \(N_ g\) from a punctured torus. \textit{J. Nielsen} [Mat. Tidsskr. B 1942, 25-41 (1942; Zbl 0027.09602)] calculated that \(h_ 0(F_ g)\), the free degree with respect to orientation-preserving homeomorphisms, is \(2g-2\) for \(g>2\) but he also showed that in the orientation-preserving setting the homeomorphism \(s\) cannot be periodic. The proof in the present paper that for no homeomorphism \(f:M\to M\) of a closed surface can \(f\), \(f^ 2,\dots,f^{h(M)}\) all be fixed point free consists of assuming about their Lefschetz numbers that \(L(f^{-2})=L(f^{-1})=L(f)=L(f^ 2)=\cdots=L(f^{h(M)})=0\) and arriving at a contradiction [compare \textit{F. B. Fuller}, Ann. M ath., II. Ser. 57, 229-230 (1953; Zbl 0050.172)].
scientific article; zbMATH DE number 62955
scientific article