Lagrangian two-spheres can be symplectically knotted.

*(English)*Zbl 1032.53068This paper shows that there are examples of a symplectic \(4\)-manifold \((M,\omega)\) with an infinite collection of Lagrangian \(2\)-spheres \(L_{i}\), all of which are differentiably isotopic but not pairwise Lagrangian isotopic.

More precisely, suppose that \((M,\omega)\) is the interior of a compact symplectic manifold with contact type boundary. Assume that \((M,\omega)\) contains Lagrangian \(2\)-spheres \(L_{1},L_{2},L_{3}\) such that \(L_{1}\cap L_{3}=\emptyset\) and \(L_{1}\cap L_{2}\) and \(L_{2}\cap L_{3}\) are transverse and consist of a single point each. Let \(\tau\) denote the Dehn twist of \(M\) along \(L_{2}\). Then, under some technical assumptions (\([\omega]=0\) and \(c_{1}(M,\omega)=0\)), the Lagrangian \(2\)-spheres \(L_{1}^{(r)} = \tau^{2r}(L_{1})\) are differentiably isotopic but not Lagrangian isotopic. They are differentiably isotopic since \(\tau^{2}\) is (differentiably) isotopic to the identity. To prove that they are not Lagrangian isotopic consists of computing the symplectic Floer cohomology groups defined in [A. Floer, J. Differ. Geom. 28, 513-547 (1988; Zbl 0674.57027)]. On the one hand \(HF(L_{1},L_{3})=0\), since they do not intersect, and on the other hand \(HF(L_{1}^{(r)},L_{3})\neq HF(L^{(s)}_1,L_3)\) for \(r\neq s\). This is computed by using a spectral sequence and a calculation of Floer cohomology groups for clean intersections [M. Poźniak, Floer homology, Novikov rings and clean intersections, Ph.D. thesis, University of Warwick, (1994); see also Am. Math. Soc. Transl., Ser. 2, Am. Math. Soc. 196(45), 119-181 (1999; Zbl 0948.57025)]. (Note that \(L_{1}^{(r)}\) and \(L_{3}\) intersect cleanly in a disjoint union of circles). The technical assumptions are necessary to deal with the grading.

The configuration of Lagrangian \(2\)-spheres \(L_{1},L_{2},L_{3}\) described above is called an \(A_{3}\)-configuration. The author also shows that there exists an \(A_{m}\)-configuration of Lagrangian \(2\)-spheres inside the (open) complex surface of \(C^{3}\) given by \(z_{1}^{2}+z_{2}^{2}+z_{3}^{m+1}+1=0\), \(|z|<R\), \(R\) large. So there are examples where the main result of the paper applies.

More precisely, suppose that \((M,\omega)\) is the interior of a compact symplectic manifold with contact type boundary. Assume that \((M,\omega)\) contains Lagrangian \(2\)-spheres \(L_{1},L_{2},L_{3}\) such that \(L_{1}\cap L_{3}=\emptyset\) and \(L_{1}\cap L_{2}\) and \(L_{2}\cap L_{3}\) are transverse and consist of a single point each. Let \(\tau\) denote the Dehn twist of \(M\) along \(L_{2}\). Then, under some technical assumptions (\([\omega]=0\) and \(c_{1}(M,\omega)=0\)), the Lagrangian \(2\)-spheres \(L_{1}^{(r)} = \tau^{2r}(L_{1})\) are differentiably isotopic but not Lagrangian isotopic. They are differentiably isotopic since \(\tau^{2}\) is (differentiably) isotopic to the identity. To prove that they are not Lagrangian isotopic consists of computing the symplectic Floer cohomology groups defined in [A. Floer, J. Differ. Geom. 28, 513-547 (1988; Zbl 0674.57027)]. On the one hand \(HF(L_{1},L_{3})=0\), since they do not intersect, and on the other hand \(HF(L_{1}^{(r)},L_{3})\neq HF(L^{(s)}_1,L_3)\) for \(r\neq s\). This is computed by using a spectral sequence and a calculation of Floer cohomology groups for clean intersections [M. Poźniak, Floer homology, Novikov rings and clean intersections, Ph.D. thesis, University of Warwick, (1994); see also Am. Math. Soc. Transl., Ser. 2, Am. Math. Soc. 196(45), 119-181 (1999; Zbl 0948.57025)]. (Note that \(L_{1}^{(r)}\) and \(L_{3}\) intersect cleanly in a disjoint union of circles). The technical assumptions are necessary to deal with the grading.

The configuration of Lagrangian \(2\)-spheres \(L_{1},L_{2},L_{3}\) described above is called an \(A_{3}\)-configuration. The author also shows that there exists an \(A_{m}\)-configuration of Lagrangian \(2\)-spheres inside the (open) complex surface of \(C^{3}\) given by \(z_{1}^{2}+z_{2}^{2}+z_{3}^{m+1}+1=0\), \(|z|<R\), \(R\) large. So there are examples where the main result of the paper applies.

Reviewer: Vicente Muñoz (Madrid)

##### MSC:

53D12 | Lagrangian submanifolds; Maslov index |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

53D40 | Symplectic aspects of Floer homology and cohomology |

57R58 | Floer homology |