##
**Holomorphic disks and three-manifold invariants: properties and applications.**
*(English)*
Zbl 1081.57013

The paper under review is a continuation of the authors’ paper [Ann. Math. 159, 1027–1158 (2004; Zbl 1073.57009)]. As their main result the authors compute the Euler number of \(HF^{\pm }(Y, s)\), relating it to known invariants such as Turaev torsion and the normalized Alexander polynomial (Theorem 1.2 and Theorem 1.8, they obtain a Künneth principle under the connected sum (Theorem 1.5) related with Thurston’s semi-norm (Theorem 1.6) and they compute the module structure of the invariants.

The paper gives two applications on complexity of three manifolds and surgeries, and bounding the number of gradient trajectories. An interesting Conjecture 1.1 is proposed concerning the relation to the Seiberg-Witten-Floer homologies.

In section 2, the authors first prove that \(\widehat{HF}(Y, s)\) and \(HF^+(Y, s)\) share the nontriviality (Proposition 2.1), and that there are finitely many Spin\(^c\) structures \(s\) such that \(HF^+(Y, s)\) is nontrivial (Theorem 2.3). Then they show that the four Floer homologies are symmetric under the involution on the Spin\(^c\) structures (Theorem 2.4), and duality under the orientation change (Proposition 2.5).

The authors give calculations for lens spaces, \(S^1 \times S^2\) and surgeries on the trefoil knot in section 3. In section 4, the authors give the equivariant Seiberg-Witten-Floer homology and computations to build the identification with the calculations done in section 3 (Proposition 4.4). It is not clear what the group \(G\) is used in the equivariant theory and the calculation does not involve the equivariant boundary maps.

The Euler characteristic \(\chi (\widehat{HF}(Y, s)) = \delta_{0, b_1(Y)}\) is given in Proposition 5.1, and Theorem 5.2 and Theorem 5.11 state that \(\chi (HF^+(Y, s))\) is the Turaev torsion function on Spin\(^c (Y)\), when \(b_1(Y) \geq 1\) and \(s\) is non-torsion. A Künneth type formula is studied in section 6 for \(\widehat{HF}, HF^{\pm }, HF^{\infty}\). The chain group under the connected sum is a tensor of chain groups, but the boundary map under the connected sum is required to justify. Theorem 1.5 (as Theorem 6.2) shows that the tensor complex is quasi-isomorphic to the complex of the connected sum (without further explicit boundary map description).

Theorem 7.1 gives the lower bound of the Thurston semi-norm by \(| \langle c_1(s), [Z] \rangle| \) for \(HF^+(Y, s)\neq 0\) and \(b_1(Y) \geq 1\), its proof relies on a construction of a special Heegaard diagram for \(Y\) containing a periodic domain representative for \(Z\). An interesting Proposition 7.5 gives a formula for \(< c_1(s), \xi >\) for a basis \(\xi \in H_2(Y, Z)\).

Section 8 is devoted to the discussion of a twisted coefficient system for the four types of Floer homologies. Theorem 9.1 gives the surgery exact sequence of \(HF^+\) for \((+ 1)\) surgery on \(Y\) and \(0\)-surgery \(Y_0\) on a knot in \(Y\). The formulation is borrowed from the instanton Floer homology exact sequence, and the proof is somehow similar and simpler. The equivariant exact sequence for Spin\(^c\) structures on the knot complement is given in Theorem 9.12 with proof sketched in section 9.2. Theorem 10.2 provides the calculation of \(HF^{\infty}\) for \(b_1(Y) \leq 2\), and the paper end with two applications described earlier.

The paper gives two applications on complexity of three manifolds and surgeries, and bounding the number of gradient trajectories. An interesting Conjecture 1.1 is proposed concerning the relation to the Seiberg-Witten-Floer homologies.

In section 2, the authors first prove that \(\widehat{HF}(Y, s)\) and \(HF^+(Y, s)\) share the nontriviality (Proposition 2.1), and that there are finitely many Spin\(^c\) structures \(s\) such that \(HF^+(Y, s)\) is nontrivial (Theorem 2.3). Then they show that the four Floer homologies are symmetric under the involution on the Spin\(^c\) structures (Theorem 2.4), and duality under the orientation change (Proposition 2.5).

The authors give calculations for lens spaces, \(S^1 \times S^2\) and surgeries on the trefoil knot in section 3. In section 4, the authors give the equivariant Seiberg-Witten-Floer homology and computations to build the identification with the calculations done in section 3 (Proposition 4.4). It is not clear what the group \(G\) is used in the equivariant theory and the calculation does not involve the equivariant boundary maps.

The Euler characteristic \(\chi (\widehat{HF}(Y, s)) = \delta_{0, b_1(Y)}\) is given in Proposition 5.1, and Theorem 5.2 and Theorem 5.11 state that \(\chi (HF^+(Y, s))\) is the Turaev torsion function on Spin\(^c (Y)\), when \(b_1(Y) \geq 1\) and \(s\) is non-torsion. A Künneth type formula is studied in section 6 for \(\widehat{HF}, HF^{\pm }, HF^{\infty}\). The chain group under the connected sum is a tensor of chain groups, but the boundary map under the connected sum is required to justify. Theorem 1.5 (as Theorem 6.2) shows that the tensor complex is quasi-isomorphic to the complex of the connected sum (without further explicit boundary map description).

Theorem 7.1 gives the lower bound of the Thurston semi-norm by \(| \langle c_1(s), [Z] \rangle| \) for \(HF^+(Y, s)\neq 0\) and \(b_1(Y) \geq 1\), its proof relies on a construction of a special Heegaard diagram for \(Y\) containing a periodic domain representative for \(Z\). An interesting Proposition 7.5 gives a formula for \(< c_1(s), \xi >\) for a basis \(\xi \in H_2(Y, Z)\).

Section 8 is devoted to the discussion of a twisted coefficient system for the four types of Floer homologies. Theorem 9.1 gives the surgery exact sequence of \(HF^+\) for \((+ 1)\) surgery on \(Y\) and \(0\)-surgery \(Y_0\) on a knot in \(Y\). The formulation is borrowed from the instanton Floer homology exact sequence, and the proof is somehow similar and simpler. The equivariant exact sequence for Spin\(^c\) structures on the knot complement is given in Theorem 9.12 with proof sketched in section 9.2. Theorem 10.2 provides the calculation of \(HF^{\infty}\) for \(b_1(Y) \leq 2\), and the paper end with two applications described earlier.

Reviewer: Weiping Li (Stillwater)

### MSC:

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

57R58 | Floer homology |

32Q65 | Pseudoholomorphic curves |