## Journal of Differential Geometry

### Localized index and $L^2$-Lefschetz fixed-point formula for orbifolds

#### Abstract

We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly, and isometrically. These localized indices, generalizing the $L^2$-index of Atiyah, are obtained by taking certain traces of the higher index for the Dirac type operators along conjugacy classes of the discrete group, subject to some trace assumption. Applying the local index technique, we also obtain an $L^2$-version of the Lefschetz fixed-point formulas for orbifolds. These cohomological formulas for the localized indices give rise to a class of refined topological invariants for the quotient orbifold.

#### Article information

Source
J. Differential Geom. Volume 102, Number 2 (2016), 285-349.

Dates
First available in Project Euclid: 27 January 2016

https://projecteuclid.org/euclid.jdg/1453910456

Digital Object Identifier
doi:10.4310/jdg/1453910456

Mathematical Reviews number (MathSciNet)
MR3454548

Zentralblatt MATH identifier
1348.58014

#### Citation

Wang, Bai-Ling; Wang, Hang. Localized index and $L^2$-Lefschetz fixed-point formula for orbifolds. J. Differential Geom. 102 (2016), no. 2, 285--349. doi:10.4310/jdg/1453910456. https://projecteuclid.org/euclid.jdg/1453910456