The finite group action and the equivariant determinant of elliptic operators III

Abstract

Let $M$ be an even-dimensional closed oriented manifold and $g$ a periodic automorphism of $M$ of order $p$. In this paper, under the assumption that the fixed points of $g^k\;(1\leq k\leq p-1)$ are isolated, a calculation formula is provided for the homomorphism $I_D:{\Bbb Z}_p\to{\Bbb R}/{\Bbb Z}$ defined in [6] for equivariant twisted signature operators $D$ over $M$. The formula gives a new method to study the periodic automorphisms of oriented manifolds. As examples of the application of the formula, results about the existence of the cyclic group action for 2,4,6-dimensional closed oriented manifolds are obtained.