##
**Holomorphic disks and topological invariants for closed three-manifolds.**
*(English)*
Zbl 1073.57009

The paper under review is to define topological invariants for closed oriented 3-manifolds \(Y\) with a Spin\(^c\) structure \(s\) via the Lagrangian Floer homology construction.

For a Heegaard surface \(\Sigma\) of \(Y\), the \(g\)-fold symmetric product \(\text{Sym}^g(\Sigma)\) is a complex manifold, where \(g\) is the genus of the surface \(\Sigma\). The \(g\)-tuples of attaching circles \(\alpha =\{\alpha_1, \dots, \alpha_g\}\) and \(\beta =\{\beta_1, \dots, \beta_g\}\) for two handlebodies give a pair of smoothly embedded \(g\)-tori \(T_{\alpha}\) and \(T_{\beta}\) in \(\text{Sym}^g(\Sigma)\).

The chain groups \(\hat{CF}(\alpha, \beta, s)\) are abelian groups freely generated by the intersection points \(T_{\alpha} \cap T_{\beta}\) with relative index defined by the Maslov index. The boundary map measures the number of \(J\)-holomorphic disks in \(\text{Sym}^g(\Sigma)\) with relative index 1 along two intersection points. Such a definition provides a chain complex for rational homology 3-spheres; by Theorem 4.1 of the paper its homology is the Floer homology \(\hat{HF}(\alpha, \beta, s)\).

Taking an integral lift on the Maslov index, one can also formulate other chain groups \(CF^{\infty}(\alpha, \beta, s)\) generated from \([x, i]\) for \(x\in T_{\alpha} \cap T_{\beta}\) and \(i\) an integer. The similar boundary map makes \(CF^{\infty}(\alpha, \beta, s)\) a chain complex and hence there is a Floer homology \(HF^{\infty}(\alpha, \beta, s)\).

The subgroup of \(CF^{\infty}(\alpha, \beta, s)\) with negative indices forms another chain subgroup \(CF^{-}(\alpha, \beta, s)\); denote its quotient group \(CF^{\infty}/CF^{-}(\alpha, \beta, s)\) by \(CF^{+}(\alpha, \beta, s)\). Hence there are four Floer homology groups. The similar chain groups with fixed orientation are defined for positive first Betti number of 3-manifolds. The paper is devoted to showing that the Floer homology groups are topological invariants, i.e. they are invariant under Heegaard moves (isotopies, handleslides, stabilizations and destabilizations of Heegaard diagrams).

In section 2, Heegaard diagrams, \(\pi_2(\text{Sym}^g(\Sigma))\), intersection points and disks and \(\text{Spin}^c\) structures are studied. The Fredholm property, transverse, compactness and orientation on the Floer setting for \((\text{Sym}^g(\Sigma), T_{\alpha}, T_{\beta})\) are discussed in section 3 (the analytic mechanism). In section 4, basic definitions of the four Floer homology groups are given.

(1) By considering some special isotopies and strongly \(s\)-admissible pointed Heegaard diagrams (in section 5), the authors show that the Floer homology groups are independent of the choice of complex structure on \(\Sigma\) and induced on \(\text{Sym}^g(\Sigma)\) (Theorem 6.1).

(2) By enlarging isotopy classes and introducing new intersections in a manageable way, Theorem 7.3 shows that the Floer homology groups are invariant under isotopy.

(3) By explicitly calculating the Floer homology groups for both Heegaard diagrams of \(\#^g (S^2\times S^1)\) and a specific handleslide, the authors in Theorem 9.5 prove the invariance property for the Floer homology groups under handleslides by using the holomorphic triangles developed in section 8.

(4) Theorem 10.1 and Theorem 10.2 provide the stabilization invariance for the Floer homology groups by a more subtle analytic gluing result for holomorphic curves.

By combining all the previous result, the authors summarize their main result inTheorem 11.1 in this paper. Further properties, computations, and applications of these Floer homology groups are given in sequels of this paper. See the authors’ paper [Ann. Math. (2) 159, No. 3, 1159–1245 (2004; Zbl 1081.57013)], for example.

For a Heegaard surface \(\Sigma\) of \(Y\), the \(g\)-fold symmetric product \(\text{Sym}^g(\Sigma)\) is a complex manifold, where \(g\) is the genus of the surface \(\Sigma\). The \(g\)-tuples of attaching circles \(\alpha =\{\alpha_1, \dots, \alpha_g\}\) and \(\beta =\{\beta_1, \dots, \beta_g\}\) for two handlebodies give a pair of smoothly embedded \(g\)-tori \(T_{\alpha}\) and \(T_{\beta}\) in \(\text{Sym}^g(\Sigma)\).

The chain groups \(\hat{CF}(\alpha, \beta, s)\) are abelian groups freely generated by the intersection points \(T_{\alpha} \cap T_{\beta}\) with relative index defined by the Maslov index. The boundary map measures the number of \(J\)-holomorphic disks in \(\text{Sym}^g(\Sigma)\) with relative index 1 along two intersection points. Such a definition provides a chain complex for rational homology 3-spheres; by Theorem 4.1 of the paper its homology is the Floer homology \(\hat{HF}(\alpha, \beta, s)\).

Taking an integral lift on the Maslov index, one can also formulate other chain groups \(CF^{\infty}(\alpha, \beta, s)\) generated from \([x, i]\) for \(x\in T_{\alpha} \cap T_{\beta}\) and \(i\) an integer. The similar boundary map makes \(CF^{\infty}(\alpha, \beta, s)\) a chain complex and hence there is a Floer homology \(HF^{\infty}(\alpha, \beta, s)\).

The subgroup of \(CF^{\infty}(\alpha, \beta, s)\) with negative indices forms another chain subgroup \(CF^{-}(\alpha, \beta, s)\); denote its quotient group \(CF^{\infty}/CF^{-}(\alpha, \beta, s)\) by \(CF^{+}(\alpha, \beta, s)\). Hence there are four Floer homology groups. The similar chain groups with fixed orientation are defined for positive first Betti number of 3-manifolds. The paper is devoted to showing that the Floer homology groups are topological invariants, i.e. they are invariant under Heegaard moves (isotopies, handleslides, stabilizations and destabilizations of Heegaard diagrams).

In section 2, Heegaard diagrams, \(\pi_2(\text{Sym}^g(\Sigma))\), intersection points and disks and \(\text{Spin}^c\) structures are studied. The Fredholm property, transverse, compactness and orientation on the Floer setting for \((\text{Sym}^g(\Sigma), T_{\alpha}, T_{\beta})\) are discussed in section 3 (the analytic mechanism). In section 4, basic definitions of the four Floer homology groups are given.

(1) By considering some special isotopies and strongly \(s\)-admissible pointed Heegaard diagrams (in section 5), the authors show that the Floer homology groups are independent of the choice of complex structure on \(\Sigma\) and induced on \(\text{Sym}^g(\Sigma)\) (Theorem 6.1).

(2) By enlarging isotopy classes and introducing new intersections in a manageable way, Theorem 7.3 shows that the Floer homology groups are invariant under isotopy.

(3) By explicitly calculating the Floer homology groups for both Heegaard diagrams of \(\#^g (S^2\times S^1)\) and a specific handleslide, the authors in Theorem 9.5 prove the invariance property for the Floer homology groups under handleslides by using the holomorphic triangles developed in section 8.

(4) Theorem 10.1 and Theorem 10.2 provide the stabilization invariance for the Floer homology groups by a more subtle analytic gluing result for holomorphic curves.

By combining all the previous result, the authors summarize their main result inTheorem 11.1 in this paper. Further properties, computations, and applications of these Floer homology groups are given in sequels of this paper. See the authors’ paper [Ann. Math. (2) 159, No. 3, 1159–1245 (2004; Zbl 1081.57013)], for example.

Reviewer: Weiping Li (Stillwater)

### MSC:

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

57R58 | Floer homology |

32Q65 | Pseudoholomorphic curves |

57R15 | Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) |