## Journal of the Mathematical Society of Japan

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

Kenji TSUBOI

#### 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.

#### Article information

Source
J. Math. Soc. Japan Volume 57, Number 1 (2005), 95-113.

Dates
First available in Project Euclid: 13 October 2006

https://projecteuclid.org/euclid.jmsj/1160745815

Digital Object Identifier
doi:10.2969/jmsj/1160745815

Mathematical Reviews number (MathSciNet)
MR2114722

Zentralblatt MATH identifier
1088.58016

#### Citation

TSUBOI, Kenji. The finite group action and the equivariant determinant of elliptic operators. J. Math. Soc. Japan 57 (2005), no. 1, 95--113. doi:10.2969/jmsj/1160745815. https://projecteuclid.org/euclid.jmsj/1160745815

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