A COINCIDENCE THEOREM FOR COMMUTING INVOLUTIONS

Let M-m be an m-dimensional, closed and smooth manifold, and let S,T : M-m -> M-m be two smooth and commuting diffeomorphisms of period 2. Suppose that S not equal T on each component of M-m. Denote by F-S and F-T the respective sets of fixed points. In this paper we prove the following coincidence theorem: if F-T is empty and the number of points of F-S is of the form 2p, with p odd, then Coinc(S,T) = {x is an element of M-m vertical bar S(x) = T(x)} has at least some component of dimension m - 1. This generalizes the classic example given by M-m = S-m, the m-dimensional sphere, S(x(0), x(1), ... , x(m)) = (-x(0), -x(1), ..., -x(m-1), x(m)) and T the antipodal map.