The signature of the fixed set of a map of odd period

Abstract:

Let $T$ be a diffeomorphism of odd period $n$ on a closed smooth manifold ${M^{2k}}$. The Conner-Floyd analysis of fixed point data and the Atiyah-Singer Index Theorem are applied to prove there exist methods of orienting the components $F$ of the fixed set of $T$, depending only on $n$, so that ${\Sigma _F}\operatorname {sgn} F \equiv \operatorname {sgn} M\bmod 4$ whenever ${T^ \ast }$ is the identity on ${H^k}(M;Q)$. Other special results of this type are obtained when assumptions are made restricting the possible eigenvalues in the normal bundle to the fixed set.