Geometry & Topology, Vol. 9(2005) Paper no. 52, pages 2261--2302.

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

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 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

AMS subject classification. Primary: 57M27. Secondary: 57R57, 53D12, 58J30.

E-print: arXiv:math.GT/0412191

DOI: 10.2140/gt.2005.9.2261

Submitted to GT on 4 December 2004. Paper accepted 1 November 2005. Paper published 6 December 2005.

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Benjamin Himpel
Mathematisches Institut, Universitaet Bonn
Beringstr. 6, D-53115 Bonn, Germany

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