Abstract
Assume that M({{mathcal {T}}}) is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph mathcal {T}. We consider the combinatorial multivariable Poincaré series associated with mathcal {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 {{mathcal {I}}} of the set of vertices) we prove surgery formulae for the normalized Seiberg–Witten invariants: the periodic constant associated with {{mathcal {I}}} appears as the difference of the Seiberg–Witten invariants of M({{mathcal {T}}}) and M({{mathcal {T}}}{setminus }{{mathcal {I}}}) for any {{mathcal {I}}}.
Highlights
Surgery formulae for 3-manifolds, focusing on certain numerical or cohomological invariant are key tools in low dimensional topology
By the new formula we present, the difference of the Seiberg–Witten invariants of two surgery manifolds is determined from a multivariable zeta–type series, which is combinatorially defined from the graph
We fix an arbitrary subset I ⊂ V of the vertices of T, and we prove that the difference of the normalized Seiberg–Witten invariants of M(T ) and M(T \I) can be computed as the periodic constant of the series with reduced variables Zh(t)|tu=1, u∈/I . (Note that the theory of quasipolynomials and the concrete computation of their periodic constants is much harder in the multivariable case.) In the case when I is the set of nodes of T we recover the ‘reduction theorem’ from [15]
Summary
Surgery formulae for 3-manifolds, focusing on certain numerical or cohomological invariant are key tools in low dimensional topology. For negative definite graph manifolds, one can formulate a different type of surgery formula, which is not imposed by purely topological theories and it has no extension (by the knowledge of the authors) to arbitrary 3-manifolds. By the new formula we present, the difference of the Seiberg–Witten invariants of two surgery manifolds is determined from a multivariable zeta–type series, which is combinatorially defined from the graph. We fix an arbitrary subset I ⊂ V of the vertices of T , and we prove that the difference of the normalized Seiberg–Witten invariants of M(T ) and M(T \I) can be computed as the periodic constant of the series with reduced variables Zh(t)|tu=1, u∈/I . 2 contains preliminaries regarding plumbing graphs, manifolds, their Seiberg–Witten invariants, and Poincaré series and their periodic constants. The last section treats the case of numerically Gorenstein graphs, where some additional nice symmetries and dualities appear
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