Journal of the Mathematical Society of Japan

The finite group action and the equivariant determinant of elliptic operators

Kenji TSUBOI

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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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1160745815

Digital Object Identifier
doi:10.2969/jmsj/1160745815

Mathematical Reviews number (MathSciNet)
MR2114722

Zentralblatt MATH identifier
1088.58016

Subjects
Primary: 58J20: Index theory and related fixed point theorems [See also 19K56, 46L80]
Secondary: 57S17: Finite transformation groups 30F99: None of the above, but in this section

Keywords
The finite group action The index theorem The equivariant determinant

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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