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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On Poincaré series associated with links of normal surface singularities


Authors: Tamás László and Zsolt Szilágyi
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 32S05, 32S25, 32S50, 57M27; Secondary 14Bxx, 32Sxx, 14J80, 57R57
DOI: https://doi.org/10.1090/tran/7802
Published electronically: June 6, 2019
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Abstract: Assume that $ M$ is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph $ \mathcal {T}$. We consider the topological Poincaré series associated with $ \mathcal {T}$ and its counting functions, which encodes rich topological information, e.g., the Seiberg-Witten invariants of $ M$.

In this article we study the counting functions via coefficient functions following the work of Szenes and Vergne. These are quasipolynomials on a special affine cone $ \mathcal {S}'$ associated with the topology of $ M$, in accordance with the previous results of Némethi and the first author. We prove that $ \mathcal {S}'$ consists of a unique quasipolynomiality chamber, and we establish further structure theorems. We provide a formula for the counting function in terms of only one- and two-variable counting functions indexed by the edges and the vertices of the graph. This is the core of the proof for a ``polynomial-negative degree part'' decomposition theorem of the Poincaré series, which leads to a polynomial generalization of the Seiberg-Witten invariants of $ M$. Finally, we reprove and discuss surgery formulas for the counting functions, in particular for the Seiberg-Witten invariants, using our methods.


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Additional Information

Tamás László
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1053 Budapest, Reáltanoda u. 13-15, Hungary
Email: laszlo.tamas@renyi.mta.hu

Zsolt Szilágyi
Affiliation: Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Kogălniceanu Street 1, 400084 Cluj-Napoca, Romania
Email: szilagyi.zsolt@math.ubbcluj.ro

DOI: https://doi.org/10.1090/tran/7802
Keywords: Normal surface singularities, links of singularities, plumbing graphs, rational homology spheres, Seiberg--Witten invariant, Poincar\'e series, coefficient functions, quasipolynomials
Received by editor(s): March 8, 2016
Received by editor(s) in revised form: January 7, 2019
Published electronically: June 6, 2019
Additional Notes: The first author was supported by NKFIH grant “Élvonal” (Frontier) KKP 126683. He was also partially supported by OTKA Grants 100796 and K112735.
The second author was supported by the ‘Lendület’ program of the Hungarian Academy of Sciences.
Article copyright: © Copyright 2019 American Mathematical Society