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

Abstract

Let T T be a diffeomorphism of odd period n n on a closed smooth manifold M 2 k {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 F of the fixed set of T T , depending only on n n , so that Σ F sgn ⁡ F ≡ sgn ⁡ M mod 4 {\Sigma _F}\operatorname {sgn} F \equiv \operatorname {sgn} M\bmod 4 whenever T ∗ {T^ \ast } is the identity on H k ( M ; Q ) {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.