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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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

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 SU(2) connections, provided M = S 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
References
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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/