A splitting formula for the spectral flow of the odd signature operator on 3–manifolds coupled to a path of SU(2) connections

Himpel, Benjamin

doi:10.2140/gt.2005.9.2261

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A splitting formula for the spectral flow of the odd signature operator on 3–manifolds coupled to a path of $SU(2)$ connections

Benjamin Himpel

Geometry & Topology 9 (2005) 2261–2302

DOI: 10.2140/gt.2005.9.2261

arXiv: math.GT/0412191

Abstract

We establish a splitting formula for the spectral flow of the odd signature operator on a closed $3$ –manifold $M$ coupled to a path of $S U (2)$ connections, provided $M = S \cup X$ , where $S$ is the solid torus. It describes the spectral flow on $M$ in terms of the spectral flow on $S$ , the spectral flow on $X$ (with certain Atiyah–Patodi–Singer boundary conditions), and two correction terms which depend only on the endpoints.

Our result improves on other splitting theorems by removing assumptions on the non-resonance level of the odd signature operator or the dimension of the kernel of the tangential operator, and allows progress towards a conjecture by Lisa Jeffrey in her work on Witten’s $3$ –manifold invariants in the context of the asymptotic expansion conjecture.

Keywords

spectral flow, odd signature operator, gauge theory, Chern–Simons theory, Atiyah–Patodi–Singer boundary conditions, Maslov index

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57R57, 53D12, 58J30

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Publication

Received: 4 December 2004

Accepted: 1 November 2005

Published: 6 December 2005

Proposed: Ronald Stern

Seconded: Peter Teichner, Ronald Fintushel

Authors

Benjamin Himpel
Mathematisches Institut Universität Bonn Beringstr. 6 D–53115 Bonn Germany
http://www.math.uni-bonn.de/people/himpel/

Volume 9, issue 4 (2005)