Algebraic & Geometric Topology

The periodic Floer homology of a Dehn twist

Michael Hutchings and Michael G Sullivan

Full-text: Open access


The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts certain embedded pseudoholomorphic curves in cross the mapping torus. It is conjectured to recover the Seiberg-Witten Floer homology of the mapping torus for most spin-c structures, and is related to a variant of contact homology. In this paper we compute the periodic Floer homology of some Dehn twists.

Article information

Algebr. Geom. Topol., Volume 5, Number 1 (2005), 301-354.

Received: 9 October 2004
Accepted: 8 March 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R58: Floer homology
Secondary: 53D40: Floer homology and cohomology, symplectic aspects 57R50: Diffeomorphisms

periodic Floer homology Dehn twist surface symplectomorphism


Hutchings, Michael; Sullivan, Michael G. The periodic Floer homology of a Dehn twist. Algebr. Geom. Topol. 5 (2005), no. 1, 301--354. doi:10.2140/agt.2005.5.301.

Export citation


  • F. Bourgeois, A Morse-Bott approach to contact homology, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001), 55–77, Fields Inst. Commun., 35, Amer. Math. Soc., Providence, RI, 2003.
  • K. Cieliebak, I. Mundet i Riera, and D. A. Salamon, Equivariant moduli problems, branched manifolds, and the Euler class, Topology 42 (2003), 641–700.
  • S. K. Donaldson, Floer homology and algebraic geometry, Vector bundles in algebraic geometry (Durham, 1993), 119–138, London Math. Soc. Lecture Note Ser., 208, Cambridge Univ Press, 1995.
  • E. Eftekhary, Floer cohomology of certain pseudo-Anosov maps on surfaces. \arxivmath.SG/0205029
  • Ya. Eliashberg, A. Givental, and H. Hofer, Ya. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. 2000, Special Volume, Part II, 560–673.
  • A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547.
  • K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), no. 5, 933–1048.
  • R. Gautschi, Floer homology of algebraically finite mapping classes, J. Symplectic Geom. 1 (2003), 715–765.
  • M. Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. 4 (2002), 313–361.
  • M. Hutchings and Y-J. Lee, Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds, Topology 38 (1999), no. 4, 861–888.
  • M. Hutchings and M. Sullivan, Rounding corners of polygons and the embedded contact homology of $T^3$. \arxivmath.SG/0410061
  • M. Hutchings and M. Thaddeus, Periodic Floer homology, in preparation.
  • S. Jabuka and T. Mark, Heegard Floer homology of certain mapping tori, Algebraic and Geometric Topology 4 (2004), 685–719.
  • Y-J. Lee, Reidemeister torsion in symplectic Floer theory and counting pseudo-holomorphic tori. \arxivmath.DG/0111313
  • D. McDuff, Singularities and positivity of intersections of $J$-holomorphic curves, pp. 191–216 in Holomorphic curves in symplectic geometry (M. Audin and F. Lafontaine, ed.), Progress in Mathematics 117, Birkhäuser, 1994.
  • P. Ozsváth and Z. Szabó, Holomorphic disks and topological invariants for closed 3-manifolds, Annals of Math. 159 (2004), 1027–1158.
  • P. Ozsváth and Z. Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Annals of Math. 159 (2004), 1159–1245.
  • P. Ozsváth and Z. Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), 58–116.
  • D. Salamon, Seiberg-Witten invariants of mapping tori, symplectic fixed points, and Lefschetz numbers, Proceedings of the 6th Gökova Geometry-Topology Conference. Turkish J. Math. 23 (1999), no. 1, 117–143.
  • M. Schwarz. Cohomology operations from $S^1$ cobordisms in Floer homology, ETH Zürich thesis, 1995.
  • P. Seidel, The symplectic Floer homology of a Dehn twist, Math. Res. Lett. 3 (1996), no. 6, 829–834.
  • P. Seidel, More about vanishing cycles and mutation, Symplectic geometry and mirror symmetry (Seoul, 2000), 429–465, World Sci. Publishing.
  • M. Sullivan, K-theoretic invariants for Lagrangian Floer homology, Geom. Funct. Anal. 12 (2002), 810–872.
  • C. H. Taubes, The Seiberg-Witten and Gromov invariants, Math. Res. Lett. 2 (1995), no. 2, 221–238.
  • C. H. Taubes, A compendium of pseudoholomorphic beasts in $\R\times (S^1\times S^2)$, Geometry and Topology 6 (2002), 657–814.
  • C. H. Taubes, Pseudoholomorphic punctured spheres in $\R\times (S^1\times S^2)$: properties and existence, preprint, 2004.