Index theory of the de Rham complex on manifolds with periodic ends

Mrowka, Tomasz; Ruberman, Daniel; Saveliev, Nikolai

doi:10.2140/agt.2014.14.3689

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Index theory of the de Rham complex on manifolds with periodic ends

Tomasz Mrowka, Daniel Ruberman and Nikolai Saveliev

Algebraic & Geometric Topology 14 (2014) 3689–3700

DOI: 10.2140/agt.2014.14.3689

Abstract

We study the de Rham complex on a smooth manifold with a periodic end modeled on an infinite cyclic cover $\tilde{X} \to X$ . The completion of this complex in exponentially weighted $L^{2}$ norms is Fredholm for all but finitely many exceptional weights determined by the eigenvalues of the covering translation map $H_{*} (\tilde{X}) \to H_{*} (\tilde{X})$ . We calculate the index of this weighted de Rham complex for all weights away from the exceptional ones.

Keywords

de Rham complex, periodic end, Alexander polynomial

Mathematical Subject Classification 2010

Primary: 58J20

Secondary: 57Q45, 58A12

References

Publication

Received: 9 February 2014

Revised: 31 July 2014

Accepted: 26 August 2014

Published: 15 January 2015

Authors

Tomasz Mrowka
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 USA

Daniel Ruberman
Department of Mathematics MS 050 Brandeis University Waltham, MA 02454 USA

Nikolai Saveliev
Department of Mathematics University of Miami PO Box 249085 Coral Gables, FL 33124 USA

Volume 14, issue 6 (2014)