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On the Fukaya categories of higher genus surfaces. (English) Zbl 1155.57029

The paper under review is to construct explicitly the Fukaya category for symplectic 2-dimensional manifolds such as oriented closed surfaces with high genus. Motived by understanding Kontsevich’s homological mirror symmetry conjecture, the Fukaya category of a symplectic manifold is still mysterious, although there is an upcoming book by Fukaya, Oh, Ohta and Ono of over a thousand pages.
The author defines a Fukaya category by taking immersed unobstructed curves as Lagrangians of the symplectic 2-manifold and defining the Floer complex of two such curves as a \(\mathbb Z_2\)-graded module over the universal Novikov ring (for convergence reasons) as the general theory. The degree is given by matching the orientations since the Lagrangians are immersed and have no proper Maslov index. The Floer differential counts the orientation preserving disks instead of holomorphic disks. Section 2 provides the details of the symplectic Floer homology of the Lagrangian curves. Then the author follows Fukaya’s \(A_{\infty}\) structure construction in this setup to define the immersed \(k\)-gons, and works on the degree in mod 2. It would be nice to understand completely the signs and index changes in the symplectic Floer theory for this simple example.
The Fukaya category is given in Section 3. Then the author provides the quasi-isomorphic Lagrangians and the cone construction from the connected sum of curves in Section 4 and 5. In order to compute the Grothendieck group of the derived category from the Fukaya category, one has to identify the zero object which is done in section 6, and to see when two Lagrangian curves represent the same class in the Grothendieck group (as given in Lemma 7.3 if they bound symplectic submanifolds).
The main theorem (Theorem 1.1) is proved in the last Section 8, showing that
\[ K_0\;({\text{Derived Fukaya category of the surface \(\Sigma\)}}) \cong H_1(S\Sigma, \mathbb Z) \oplus \mathbb R, \]
where \(S\Sigma\) is the unit tangent bundle of the surface. The proof follows from the construction of the class of the boundary of a cylinder and the class of a curve bounding a torus of zero area in the Grothendieck group. The surjectivity follows from the holonomy and winding number properties as in Section 7, and injectivity from Lickorish generators of the mapping class group and the transitivity of the mapping class group acting on non-separating curves.
The paper provides a good introduction to Fukaya category with a slightly different construction (essentially a combinatorial construction) in the well written appendices.

MSC:

57R58 Floer homology
53D40 Symplectic aspects of Floer homology and cohomology
53D12 Lagrangian submanifolds; Maslov index
57R17 Symplectic and contact topology in high or arbitrary dimension
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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References:

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