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Lagrangian two-spheres can be symplectically knotted. (English) Zbl 1032.53068
This 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.

##### 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
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