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Holomorphic disks and genus bounds. (English) Zbl 1056.57020

One of the important results using Seiberg-Witten theory is Kronheimer and Mrowka’s proof of the Thom conjecture (an algebraic curve in \(\mathbb{C} P^2\) realizes the minimal genus in a given homology class). The 3-dimensional version of the Thom conjecture was studied by D. Auckley [Osaka J. Math. 33, No. 3, 737–750 (1996; Zbl 0881.57034)]. Auckley proved that the Thurston norm \(\Theta (\alpha)\) on \(H_2(Y; \mathbb{Z})\) for irreducible atoridal 3-manifolds \(Y\) satisfies \(\Theta (\alpha ) \geq | c_1(s) \cap F_{\alpha}| \) provided the algebraic number \(\lambda (Y, s)\) of Seiberg-Witten solutions on \(Y\) is nonzero, where \(s\) is a spin\(^c\)- structure and \(F_{\alpha}\) is an embedded surface realizing \(2g(F_{\alpha})-2 = \Theta (\alpha)\) and \([F_{\alpha}] = \alpha \in H_2(Y; \mathbb{Z})\). It follows that for every \(\alpha \in H_2(Y; \mathbb{Z})\), \[ \Theta (\alpha ) \geq \max_{s\in \text{Spin}^c(Y), \lambda (Y, s)\neq 0} | \langle c_1(s), \alpha \rangle| . \] In the paper under review, the authors prove the following inequality for \(\alpha \in H_2(Y; \mathbb{Z})\) \[ \Theta (\alpha ) \geq \max_{s\in \text{Spin}^c(Y), \underline{\widehat{HF}} (Y, s) \neq 0} | \langle c_1(s), \alpha \rangle| , \] where the hypothesis is stronger via the nontrivial Heegaard Floer homology \(\underline{\widehat{HF}}(Y, s)\). The first ingredient to prove the result is Gabai’s result that there is a smooth, taut foliation \({\mathcal F}\) which contains \(F_{\alpha}\) as a union of compact leaves under a proper assumption on \(Y\). This implies that \(c_1({\mathcal F})\cap [F_{\alpha}] = \Theta (\alpha)\), the left hand side of the inequality. By the existence of such a smooth taut foliation, Eliashberg and Thurston showed that \([-1, 1]\times Y\) can be equipped with a convex symplectic form which extends the foliation \({\mathcal F}\) and provides a weakly symplectically semi-fillable contact structure \(\xi\) with \(c_1(\xi) \cap F_{\alpha} = \Theta (\alpha)\). This corresponds to a spin\(^c\)-structure on \(Y\) with a nontrivial Heegaard Floer homology class. The non-triviality is analogous to a non-vanishing theorem of Seiberg-Witten invariants on symplectic 4-manifolds by Taubes’ work. Hence the identity holds. Section 1 gives the introduction and states the main result, Theorem 1.1, and its consequence. Section 2 provides the contact geometry background. A review of variants of Heegaard Floer homologies and their properties is given in section 3. The construction of the Heegaard Floer homology class associated to a contact structure is given in section 4. The last section 5 completes the proof of Theorem 1.1 and several corollaries. Note that with Gabai’s and Eliashberg-Thurston’s results, Auckley’s result can be improved to \[ \Theta (\alpha ) = \max_{s\in \text{Spin}^c(Y), \lambda (Y, s)\neq 0} | \langle c_1(s), \alpha \rangle| . \]

MSC:

57R58 Floer homology
53D40 Symplectic aspects of Floer homology and cohomology
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57R95 Realizing cycles by submanifolds

Citations:

Zbl 0881.57034
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