Noncoincidence index, free group actions, and the fixed point property for manifolds

Abstract

Let M be a compact oriented connected topological manifold. We show that if the Euler characteristic χ{M) Φ 0 and M admits no degree zero self-maps without fixed points, then there is a finite number r such that any set of r or more fixed-point-free self-maps of M has a coincidence (i.e. for two of the maps / and g there exists x e M so that f(x) - g(x)). We call r the noncoincidence index of M. More generally, for any manifold M with χ(M) Φ 0 there is a finite number r (called the restricted noncoincidence index of M) so that any set of r or more fixed-point-free nonzero degree self-maps of M has a coincidence. We investigate how these indices change as one passes from a space to its orbit space under a free action. We compute the restricted noncoincidence index for certain products and for the homogeneous spaces SUn/K, K a closed connected subgroup of maximal rank; in some cases these computations also give the noncoincidence index of the space.