#### Volume 14, issue 6 (2014)

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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
##### Abstract

We study the de Rham complex on a smooth manifold with a periodic end modeled on an infinite cyclic cover $\stackrel{̃}{X}\to X$. 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 ${H}_{\ast }\left(\stackrel{̃}{X}\right)\to {H}_{\ast }\left(\stackrel{̃}{X}\right)$. 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
##### 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