##
**Monopoles and lens space surgeries.**
*(English)*
Zbl 1204.57038

The paper under review is to prove that real projective three space cannot be obtained from Dehn surgery on a nontrivial knot in the three sphere by the monopole Floer homology, and to prove that monopole Floer homology detects the unknot among other things.

The main result of the paper is Theorem 1.1 stating that if there is an orientation-preserving diffeomorphism \(S^3_r(K) \cong S^3_r(U)\) for some \(r\in \mathbb Q\), then \(K = U\) the unknot, where \(S^3_r(K)\) is the resulting manifold obtained from \(r\)-Dehn surgery on the knot \(K\) in \(S^3\). The cases \(r=0\), \(r\) nonintegral, \(r= \pm 1\) are known by Gabai, Culler-Gordon-Luecke-Shalen and Gordon-Luecke respectively. The main ingredients in the proof are the nonvanishing result for the monopole Floer homology of 3-manifolds with taut foliations and the surgery exact sequence.

The authors list several formal properties for the monopole Floer homology in section 2, explain the (canonical) gradings, completions, reducible contributions and the role of Spin\(^c\) structures. In section 3, by fixing the 2-plane fields and orientation, the monopole Floer homology groups of \(S^3\) and \(S^3_p(U)\) for positive integers \(p\) are calculated. Then the authors define \(p\)-standard knots and show that an orientation-diffeomorphism between \(S^3_p(K)\) and \(S^3_p(U)\) for some positive integer \(p\) implies the knot is \(p\)-standard, and \(0\)-standard via the surgery exact sequences and induction on \(p\).

The authors give the basic construction and chain complexes of monopole Floer homology in section 4 without proofs, refering to the first two authors’ upcoming book. Section 5 is devoted to one of the key ingredients in the proof of Theorem 1.1, i.e., to prove that the surgery exact sequence for monopole Floer homology holds. The method is similar to the one used by Floer for the instanton Floer homology via different decompositions of cobordisms and chain homotopy maps induced from them. Section 6 gives an important criterion for the nonvanishing theorem for 3-manifolds admitting taut foliations and not \(S^1\times S^2\). The image \(j_*\) from HM-to to HM-from of \(S^3_0(K)\) with local coefficients is nonzero; and the image of \(j_*\) is zero for the unknot.

The proof is based on results by the first two authors in their monopoles and contact structures paper. Section 7 is to re-prove the case covered by Culler, Gordon, Luecke and Shalen by the monopole Floer theory and Gabai’s construction. Section 8 contains several applications of the techniques developed in this paper. (1) for a knot \(K\) in \(S^3\) with \(S^3_p(K)\) a lens space for some integer \(p\), the Seifert genus of the knot \(K\) is the degree of the symmetrized Alexander polynomial of \(K\); (2) if \(|p|<9\) in (1), then \(K\) is either the unknot or the trefoil; (3) there is no rational number \(r\geq 0\) such that \(S_r^3(K)\) is a positively oriented Seifert fibered space if the degree of the Alexander polynomial is strictly less than its Seifert genus. The last section discusses three manifolds admitting no taut foliations and defines a monopole \(L\)-space. By the nonvanishing results, a monopole \(L\)-space admits no taut foliations. There are other applications in this paper.

The main result of the paper is Theorem 1.1 stating that if there is an orientation-preserving diffeomorphism \(S^3_r(K) \cong S^3_r(U)\) for some \(r\in \mathbb Q\), then \(K = U\) the unknot, where \(S^3_r(K)\) is the resulting manifold obtained from \(r\)-Dehn surgery on the knot \(K\) in \(S^3\). The cases \(r=0\), \(r\) nonintegral, \(r= \pm 1\) are known by Gabai, Culler-Gordon-Luecke-Shalen and Gordon-Luecke respectively. The main ingredients in the proof are the nonvanishing result for the monopole Floer homology of 3-manifolds with taut foliations and the surgery exact sequence.

The authors list several formal properties for the monopole Floer homology in section 2, explain the (canonical) gradings, completions, reducible contributions and the role of Spin\(^c\) structures. In section 3, by fixing the 2-plane fields and orientation, the monopole Floer homology groups of \(S^3\) and \(S^3_p(U)\) for positive integers \(p\) are calculated. Then the authors define \(p\)-standard knots and show that an orientation-diffeomorphism between \(S^3_p(K)\) and \(S^3_p(U)\) for some positive integer \(p\) implies the knot is \(p\)-standard, and \(0\)-standard via the surgery exact sequences and induction on \(p\).

The authors give the basic construction and chain complexes of monopole Floer homology in section 4 without proofs, refering to the first two authors’ upcoming book. Section 5 is devoted to one of the key ingredients in the proof of Theorem 1.1, i.e., to prove that the surgery exact sequence for monopole Floer homology holds. The method is similar to the one used by Floer for the instanton Floer homology via different decompositions of cobordisms and chain homotopy maps induced from them. Section 6 gives an important criterion for the nonvanishing theorem for 3-manifolds admitting taut foliations and not \(S^1\times S^2\). The image \(j_*\) from HM-to to HM-from of \(S^3_0(K)\) with local coefficients is nonzero; and the image of \(j_*\) is zero for the unknot.

The proof is based on results by the first two authors in their monopoles and contact structures paper. Section 7 is to re-prove the case covered by Culler, Gordon, Luecke and Shalen by the monopole Floer theory and Gabai’s construction. Section 8 contains several applications of the techniques developed in this paper. (1) for a knot \(K\) in \(S^3\) with \(S^3_p(K)\) a lens space for some integer \(p\), the Seifert genus of the knot \(K\) is the degree of the symmetrized Alexander polynomial of \(K\); (2) if \(|p|<9\) in (1), then \(K\) is either the unknot or the trefoil; (3) there is no rational number \(r\geq 0\) such that \(S_r^3(K)\) is a positively oriented Seifert fibered space if the degree of the Alexander polynomial is strictly less than its Seifert genus. The last section discusses three manifolds admitting no taut foliations and defines a monopole \(L\)-space. By the nonvanishing results, a monopole \(L\)-space admits no taut foliations. There are other applications in this paper.

Reviewer: Weiping Li (Stillwater)

### MSC:

57R58 | Floer homology |

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

57R57 | Applications of global analysis to structures on manifolds |