Heegaard Floer homology of certain mapping tori. (English) Zbl 1052.57046

The authors calculate the Heegaard Floer homologies \(HF^{+}(M,s)\) for mapping tori \(M\) associated to certain surface diffeomorphisms, where \(s\) is any Spin\(^{c}\) structure on \(M\) whose first Chern class is non-torsion. The examples they consider are the mapping tori of the diffeomorphisms \(t_{\gamma}^{m} \circ t_{\delta}^{n}\) for \(m,n \in \mathbb{Z}\) and that of \(t_{\sigma}^{+-1}\), where \( t_{\gamma}, t_{\delta},t_{\sigma}\) are the right-handed Dehn twists about a pair \(\gamma, \delta \) of geometrically dual nonseparating curves on a genus \(g\) Riemann surface \(\Sigma_{g}\), and a curve \(\sigma\) separating \(\Sigma_{g}\) into components of genus 1 and \(g-1\).


57R58 Floer homology
53D40 Symplectic aspects of Floer homology and cohomology
Full Text: DOI arXiv EuDML EMIS


[1] E Eftekhary, Floer homology of certain pseudo-Anosov maps, J. Symplectic Geom. 2 (2004) 357 · Zbl 1081.53075
[2] R Fintushel, R J Stern, Using Floer’s exact triangle to compute Donaldson invariants, Progr. Math. 133, Birkhäuser (1995) 435 · Zbl 0844.57017
[3] M Hutchings, M Sullivan, The periodic Floer homology of a Dehn twist, Algebr. Geom. Topol. 5 (2005) 301 · Zbl 1089.57021
[4] I G Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962) 319 · Zbl 0121.38003
[5] C T McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. \((4)\) 35 (2002) 153 · Zbl 1009.57021
[6] G Meng, C H Taubes, SW=Milnor torsion, Math. Res. Lett. 3 (1996) 661 · Zbl 0870.57018
[7] D A Neumann, 3-manifolds fibering over \(S^1\), Proc. Amer. Math. Soc. 58 (1976) 353 · Zbl 0334.57003
[8] P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179 · Zbl 1025.57016
[9] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58 · Zbl 1062.57019
[10] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. \((2)\) 159 (2004) 1159 · Zbl 1081.57013
[11] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. \((2)\) 159 (2004) 1027 · Zbl 1073.57009
[12] P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326 · Zbl 1099.53058
[13] P Ozsváth, Z Szabó, Holomorphic triangle invariants and the topology of symplectic four-manifolds, Duke Math. J. 121 (2004) 1 · Zbl 1059.57018
[14] P Seidel, The symplectic Floer homology of a Dehn twist, Math. Res. Lett. 3 (1996) 829 · Zbl 0876.57022
[15] W P Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 59 (1986) · Zbl 0585.57006
[16] J L Tollefson, 3-manifolds fibering over \(S^1\) with nonunique connected fiber, Proc. Amer. Math. Soc. 21 (1969) 79 · Zbl 0175.50003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.