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 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

Primary: 57M27

Secondary: 53D12, 57R57, 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