The Seiberg-Witten invariants of negative definite plumbed 3-manifolds. (English) Zbl 1231.32020
The topology of a normal complex surface singularity is determined by the resolution graph \(\Gamma\). In fact, a small neighborhood of the singularity is homeomorphic to the cone over the link \(M\) and \(M\) is a plumbed 3-manifold associated with \(\Gamma\).
Suppose that the plumbed 3-manifold \(M\) is a rational homology sphere. In the paper under review, the author studies the Seiberg-Witten invariant of the manifold \(M\) in view of invariants of surface singularities, and provides two new combinatorial formulas for the Seiberg-Witten invariant. The first one is sort of the constant term of a “multivariable Hilbert polynomial”. More precisely, it is the periodic constant of a zeta function \(Z(\mathbf t)\) associated with the graph \(\Gamma\). The periodic constant plays an important role in the addition formula for the geometric genus of splice quotients [T. Okuma, Trans. Am. Math. Soc. 360, No. 12, 6643–6659 (2008; Zbl 1162.32017)] and a surgery formula for the Seiberg-Witten invariant given by G. Braun and the author [J. Reine Angew. Math. 638, 189–208 (2010; Zbl 1232.57013)]. As these formulas implied the Seiberg-Witten invariant conjecture of the author and L. I. Nicolaescu for splice quotients [Geom. Topol. 6, 269–328 (2002; Zbl 1031.32023)], the first formula in the present paper verifies that the identity of \(Z(\mathbf t)\) with a series determined by the “divisorial multi-index filtration” implies the conjecture.
The second formula realizes the Seiberg-Witten invariant as the normalized Euler characteristic of the lattice cohomology associated with \(\Gamma\) [the author, Publ. Res. Inst. Math. Sci. 44, No. 2, 507–543 (2008; Zbl 1149.14029)]. This theory is considered as a highly sophisticated generalization of a method using Laufer type computation sequences. The formula supports the conjectural connections between the Seiberg-Witten-Floer homology, or the Heegaard-Floer homology, and the lattice cohomology.
Suppose that the plumbed 3-manifold \(M\) is a rational homology sphere. In the paper under review, the author studies the Seiberg-Witten invariant of the manifold \(M\) in view of invariants of surface singularities, and provides two new combinatorial formulas for the Seiberg-Witten invariant. The first one is sort of the constant term of a “multivariable Hilbert polynomial”. More precisely, it is the periodic constant of a zeta function \(Z(\mathbf t)\) associated with the graph \(\Gamma\). The periodic constant plays an important role in the addition formula for the geometric genus of splice quotients [T. Okuma, Trans. Am. Math. Soc. 360, No. 12, 6643–6659 (2008; Zbl 1162.32017)] and a surgery formula for the Seiberg-Witten invariant given by G. Braun and the author [J. Reine Angew. Math. 638, 189–208 (2010; Zbl 1232.57013)]. As these formulas implied the Seiberg-Witten invariant conjecture of the author and L. I. Nicolaescu for splice quotients [Geom. Topol. 6, 269–328 (2002; Zbl 1031.32023)], the first formula in the present paper verifies that the identity of \(Z(\mathbf t)\) with a series determined by the “divisorial multi-index filtration” implies the conjecture.
The second formula realizes the Seiberg-Witten invariant as the normalized Euler characteristic of the lattice cohomology associated with \(\Gamma\) [the author, Publ. Res. Inst. Math. Sci. 44, No. 2, 507–543 (2008; Zbl 1149.14029)]. This theory is considered as a highly sophisticated generalization of a method using Laufer type computation sequences. The formula supports the conjectural connections between the Seiberg-Witten-Floer homology, or the Heegaard-Floer homology, and the lattice cohomology.
Reviewer: Tomohiro Okuma (Yamagata)
MSC:
| 32S25 | Complex surface and hypersurface singularities |
| 32S05 | Local complex singularities |
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 32S50 | Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants |
| 32C35 | Analytic sheaves and cohomology groups |
| 57R57 | Applications of global analysis to structures on manifolds |
Keywords:
normal surface singularities; links of singularities; Seiberg-Witten invariants; Hilbert polynomials; Heegaard-Floer homologyReferences:
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