Let M be a compact oriented
connected topological manifold. We show that if the Euler characteristic χ(M)≠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 f and g there exists x ∈ M so that
f(x) = g(x)). We call r the noncoincidence index of M. More generally, for any
manifold M with χ(M)≠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.
