#### Volume 9, issue 4 (2005)

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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
 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 $SU\left(2\right)$ 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