The finite group action and the equivariant determinant of elliptic operators

Abstract

If a closed oriented manifold admits an action of a finite group $G$, the equivariant determinant of a $G$-equivariant elliptic operator on the manifold defines a group homomorphism from $G$ to $S^1$. The equivariant determinant is obtained from the fixed point data of the action by using the Atiyah-Singer index theorem, and the fact that the equivariant determinant is a group homomorphism imposes conditions on the fixed point data. In this paper, using the equivariant determinant, we introduce an obstruction to the existence of a finite group action on the manifold, which is obtained directly from the relation among the generators of the finite group.