ABSTRACT. A differentiable orientation preserving map of odd prime period on a closed oriented differentiable manifold gives rise to two invariants taking values in a Witt group of bilinear forms. One is globally defined in terms of the rational cohomology of the manifold and the other is locally defined in terms of the fixed point set and its normal bundle. We show that these two invariants are, in fact, equal and apply this result to relate the structure of the manifold to that of the fixed point set and the quotient space.
Let T be a diffeomorphism of odd period n on a closed smooth manifold M. 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 «, so that 2f sgn F m sgn Ai mod 4 whenever T* is the identity on Hk(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.
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