The problem of whether a
manifold M admits a fixed point free map is an old one. One well known result is
that if the Euler characteristic χ(M) = 0, then M has such a map. In the case where
M is a closed differentiable manifold this follows easily from the fact that χ(M) = 0
if and only if the tangent bundle of M admits a nonzero cross-section (see Hopf [4]).
But χ(S2n) = 2, and S2n certainly admits a fixed point free map, namely, the
antipodal map. Therefore, the vanishing of the Euler characteristic of a manifold is
only a sufficient, though hardly a necessary, condition for the manifold to have
a fixed point free map. In the search for other invariants it is natural to
generalize somewhat and state the problem in terms finding coincidence free
maps.
The object of this paper is to give an elementary proof of the fact that,
given a continuous map f : (Wn,δWn) → (Mn,∂Mn) between oriented
C∞-manifolds, there is a well defind obstruction o(f) to finding a special sort of map
F : M → M with the property that F(x)≠f(x) for all x ∈ W. This is the content
of Theorem 1 in §2. F will not necessarily be homotopic to f, but then
this is something that should not be required in view of the fact that the
antipodal map on S2n is not homotopic to the identity map either. In Theorem
2 we prove that o(identity) behaves naturally with respect to tangential
maps.
