## 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 $\Theta (\alpha ) = \max_{s\in \text{Spin}^c(Y), \lambda (Y, s)\neq 0} | \langle c_1(s), \alpha \rangle| .$

### MathOverflow Questions:

On the genus of thin knots and the degree of the Alexander polynomial

### 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

Zbl 0881.57034
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