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The periodic Floer homology of a Dehn twist. (English) Zbl 1089.57021

For a compact surface \(\Sigma\) with a symplectic form \(\omega\) and \(\phi\) a symplectic diffeomorphism of \((\Sigma, \omega)\), one can identify the periodic orbits of \(\phi \) with the closed orbits of the natural flow \(R_{\phi }\) in the mapping torus \(Y_{\phi }\). In order to define a homology theory, one proceeds as follows. The generators of the chain complex are admissible orbit sets over \(\mathbb Z_2\). The differential \(\delta \) is counting the evenness or oddness of the one-dimensional flow lines (\(J\)-holomorphic curves) from one admissible orbit to another one in the relative homology class. It is supposed that \(\delta^2 =0\); details of the construction are said to be published in the first author’s joint paper with M. Thaddeus, “Periodic Floer Homology”, which was not available at the time this review was written. Once this is done, the homology of the chain complex is the period Floer homology (PFH for short) of \((\phi, h)\).
The paper under review is to compute the PFH of some Dehn twists under the assumption that \(\delta^2 = 0\) is true.
If the degree \(d\) (defined to be the intersection number of \(h\in H_1(Y_{\phi})\) with a fiber of the mapping torus \(Y_{\phi} \to \mathbb R/\mathbb Z\)) equals one, this is the homology of a chain complex generated by the fixed points of \(\phi\) and counting the pseudo-holomorphic sections of \(\mathbb R \times Y_{\phi} \to \mathbb R \times S^1\). The symplectic Floer homology has been computed for a Dehn twist by Seidel.
In Theorem 3.5 of the paper, the authors obtain a combinatorial formula for most of the differentials in the chain complex in terms of rounding corners of convex polygonal paths connecting lattice points in the plane. One of the main results is Theorem 4.1 which computes the PFH of an \(n\)-fold positive Dehn twist on the torus; Theorem 5.3 and 5.4 compute the periodic Floer homology of Dehn twists along distinct circles on the surface \(\Sigma \) with \(\phi\) close to the identity away from the circles.
Section 2 reviews the basic notions and constructions of the PFH, a basic duality property and a relative index inequality. Section 3 gives the PFH of a twist on a cylinder in Theorem 3.1. The authors identify the index (Proposition 3.2) and the differential (Theorem 3.5) explicitly in this case.
One of the main results is Theorem 4.1 on the computation of the PFH of an \(n\)-parallel positive Dehn twist on the torus. The last section 5 is devoted to the computation of the PFH for a nonseparating positive Dehn twist on a closed surface \(\Sigma\) with its genus \(g_{\Sigma } \geq 2 d +1\) in Theorem 5.3, and the computation for a separating positive Dehn twist on a surface in Theorem 5.4.
There are too many Floer homologies on 3-manifolds from the symplectic, instanton and monopole point of view (see the motivation of the paper). It is possible and has been stated in various conjectures that all these Floer homologies are equivalent to each other. It would be nice to see if this is indeed the case. But the authors did not yet go further on in this direction.

MSC:

57R58 Floer homology
53D40 Symplectic aspects of Floer homology and cohomology
57R50 Differential topological aspects of diffeomorphisms
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References:

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