Minimizing the number of fixed points for self-maps of compact surfaces

Abstract

Let P denote the topologcal 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 / of P. We present here a formula for the value of MF(f). Note that MF(f) depends on the induced homomorphism, / # , on fundamental group, so our formula concerns the two relevant words in the free group on the letters a and b corresponding to the loops which comprise the figure eight. Special case: Let g m : P - P, m > 0, be given such that (g m ) # (fl) -{bab- a~) m ba 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.