Splitting the spectral flow and the Alexander matrix (Q1343324)

This paper studies the problem of computing the spectral flow \(\text{SF} (\alpha, \beta)\) of the Atiyah-Patodi-Singer operator between two flat SU(2)-connexions \(\alpha, \beta\) on a 3-manifold \(Z\), when \(Z\) is split along a torus so that \(\alpha,\beta\) can be connected by a path of flat connexions on each piece. A well-known theorem of Yoshida shows that, if there are no singularities on the paths, \(\text{SF} (\alpha, \beta)\) only depends, in a simple explicit way, on the restriction of the paths to the splitting torus. This result has now been generalized by Cappell-Lee-Miller. In the present paper this problem is treated in cases where the pieces bound 4-manifolds (after filling in the torus with a solid torus) over which \(\alpha, \beta\) extend, by use of the Atiyah-Patodi-Singer Index theorem. In one application \(Z\) is a surgery on a satellite of a knot \(K\) and \(\alpha, \beta\) are reducible on the complement of a tubular neighborhood of \(K\). Then \(\alpha, \beta\) induce flat connexions on \(Z_0\), the manifold obtained by surgery along the same satellite of the trivial knot. The difference between the values of \(\text{SF} (\alpha, \beta)\) on \(Z\) and \(Z_0\) is computed and the result is a formula in terms of the (twisted) knot signatures of \(K\) defined by \(\alpha\) and \(\beta\) and the multiplicities of the corresponding roots of the Alexander polynomial. Since the restrictions to the torus are the same in both cases, this shows that Yoshida's formula requires a correction term when there are singularities. The formula of Cappell-Lee-Miller supplies such a correction in a more abstract context. Another application is to the case of graph manifolds where the two pieces of \(Z\) are Seifert fibred homology knot complements and uses methods of Fintushel-Stern.