Vol. 6, No. 3, 2021

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The relative $L^2$ index theorem for Galois coverings

Moulay-Tahar Benameur

Vol. 6 (2021), No. 3, 503–541
Abstract

Given a Galois covering of complete spin manifolds where the base metric has positive scalar curvature near infinity, we prove that for small enough 𝜖 > 0, the 𝜖 spectral projection of the Dirac operator has finite trace in the Atiyah von Neumann algebra. This allows us to define the L2 index in the even case, and we prove its compatibility with the Xie–Yu higher index and deduce L2 versions of the classical Gromov–Lawson relative index theorems. Finally, we briefly discuss some Gromov–Lawson L2 invariants.

Keywords
$L^2$ index theorem, Galois covering, relative index, Gromov–Lawson theorem, Dirac operator, complete manifold, spin structure, positive scalar curvature
Mathematical Subject Classification 2010
Primary: 32Q10, 53C27
Secondary: 53C12, 57R30
Milestones
Received: 3 June 2020
Revised: 7 October 2020
Accepted: 4 January 2021
Published: 11 September 2021
Authors
Moulay-Tahar Benameur
Institut Montpellierain Alexander Grothendieck
Université de Montpellier
Montpellier
France