Minimizing the number of fixed points for self-maps of compact surfaces
Let P denote the topological space obtained by taking a closed regular neighborhood of the figure-eight in the plane. Let MF(f) denote the minimum number of fixed points achievable among maps homotopic to a given self-map f of P. We present here a formula for the value of MF(f). Note that MF(f) depends on the induced homomorphism, \(f_{\#}\), on fundamental group, so our formula concerns the two relevant words in the free group on the lettes a and b corresponding to the loops which comprise the figure eight. Special case: Let \(g_ m: P\to P\), \(m\geq 0\), be given such that \((g_ m)_{\#}(a)=(bab^{-1}a^{-1})^ mba\) and \((g_ m)_{\#}(b)=1\). It is easy to show that the Nielsen number of \(g_ m\), \(N(g_ m)\), is equal to zero. On the other hand, our formula shows that \(MF(g_ m)=2m\). Hence the difference between N(f) and MF(f) can be made arbitrarily large.
regular neighborhood of the figure-eight in the plane
scientific article; zbMATH DE number 3911283