Surgery formulae for the Seiberg-Witten invariant of plumbed 3-manifolds.

Abstract

Assume that $M\left(T\right)$ is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph $T$ . We consider the combinatorial multivariable Poincaré series associated with $T$ and its counting functions, which encode rich topological information. Using the 'periodic constant' of the series (with reduced variables associated with an arbitrary subset $I$ of the set of vertices) we prove surgery formulae for the normalized Seiberg-Witten invariants: the periodic constant associated with $I$ appears as the difference of the Seiberg-Witten invariants of $M\left(T\right)$ and $M(T\backslash I)$ for any $I$ .