Surgery formula for Seiberg–Witten invariants of negative definite plumbed 3-manifolds

Abstract

We derive a cut-and-paste surgery formula of Seiberg-Witten in- variants for negative definite plumbed rational homology 3-spheres. It is sim- ilar to (and motivated by) Okuma's recursion formula (27, 4.5) targeting an- alytic invariants of splice-quotient singularities. Combining the two formulas automatically provides a proof of the equivariant version (11, 5.2(b)) of the Seiberg-Witten invariant conjecture (18) for these singularities. Problem 5 of the review article (30) of Ozsvath and Szabo is to develop cut-and- paste techniques for calculating the Heegaard Floer homology of 3-manifolds. In this article we obtain a possible answer at the level of the Seiberg-Witten invariant (i.e. at the level of the normalized Euler characteristic of the Heegaard Floer homol- ogy): we provide the cut-and-paste surgery formula (1.0.3) for the Seiberg-Witten invariants of plumbed rational homology 3-spheres associated with negative definite plumbing graphs. In order to state it, we fix some notations (for more details, see §3). For any graph G, let V(G) denote its set of vertices. Let |S| denote the size of the finite set S. Thus, |V(G)| is the number of vertices of G. Let be a connected plumbing graph. Each vertex w ∈ V() is decorated by an integer bw. Let e X() be the 4-manifold with boundary obtained by plumbing from , which we briefly recall. The manifold e X() is a tubular neighbourhood of oriented 2-spheres Ew associated with the vertices w of the graph. For every two adjacent vertices, their 2-spheres intersect transversally at one point; beside these, the 2-spheres do not intersect each other. The number bw is the Euler number of the normal bundle of the 2-sphere of the vertex w. The manifold e