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 letters a and b corresponding to the loops which comprise the figure eight.
Special case: Let gm: P → P, m ≥ 0, be given such that (gm)#(a) = (bab−1a−1)mba
and (gm)#(b) = 1. It is easy to show that the Nielsen number of gm, N(gm), is
equal to zero. On the other hand, our formula shows that MF(gm) = 2m.
Hence the difference between N(f) and MF(f) can be made arbitrarily
large.
