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An index theorem for end-periodic operators

Part of: Partial differential equations on manifolds; differential operators Parabolic equations and systems

Published online by Cambridge University Press:  07 September 2015

Tomasz Mrowka ,
Daniel Ruberman  and
Nikolai Saveliev
Tomasz Mrowka
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA email mrowka@mit.edu
Daniel Ruberman
Affiliation:
Department of Mathematics, MS 050, Brandeis University, Waltham, MA 02454, USA email ruberman@brandeis.edu
Nikolai Saveliev
Affiliation:
Department of Mathematics, University of Miami, PO Box 249085, Coral Gables, FL 33124, USA email saveliev@math.miami.edu
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Abstract

We extend the Atiyah, Patodi, and Singer index theorem for first-order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes’ Fredholm theory for general end-periodic operators. Our index theorem is expressed in terms of a new periodic eta-invariant that equals the Atiyah–Patodi–Singer eta-invariant in the cylindrical setting. We apply this periodic eta-invariant to the study of moduli spaces of Riemannian metrics of positive scalar curvature.

Keywords

index theoremFredholm operatorperiodic endpositive scalar curvature

MSC classification

Primary: 58J20: Index theory and related fixed point theorems 58J28: Eta-invariants, Chern-Simons invariants 58J35: Heat and other parabolic equation methods 35K05: Heat equation

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 2 , February 2016 , pp. 399 - 444
Copyright
© The Authors 2015 

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