A coincidence theorem for commuting involutions

Abstract

Let M m M^{m} be an m m -dimensional, closed and smooth manifold, and let S , T : M m → M m S, T:M^{m} \to M^{m} be two smooth and commuting diffeomorphisms of period 2 2 . Suppose that S ≠ T S \not = T on each component of M m M^{m} . Denote by F S F_{S} and F T F_{T} the respective sets of fixed points. In this paper we prove the following coincidence theorem: if F T F_{T} is empty and the number of points of F S F_{S} is of the form 2 p 2p , with p p odd, then C o i n c ( S , T ) = { x ∈ M m | S ( x ) = T ( x ) } Coinc(S,T)=\{x \in M^{m} \ \vert \ S(x)=T(x) \} has at least some component of dimension m − 1 m-1 . This generalizes the classic example given by M m = S m M^{m}=S^{m} , the m m -dimensional sphere, S ( x 0 , x 1 , . . . , x m ) = ( − x 0 , − x 1 , . . . , − x m − 1 , x m ) S(x_{0},x_{1},...,x_{m}) = (-x_{0},-x_{1},...,-x_{m-1},x_{m}) and T T the antipodal map.