Minimizing the number of fixed points for self-maps of compact surfaces
Michael R. Kelly · Pacific Journal of Mathematics · 1987
Let P denote the topologίcal 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-ι a~ι) 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.