#### 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.

Notes on file formats
Benjamin Himpel

Mathematisches Institut, Universitaet Bonn

Beringstr. 6, D-53115 Bonn, Germany

Email: himpel@math.uni-bonn.de

URL: http://www.math.uni-bonn.de/people/himpel/

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