Algebraic & Geometric Topology

The Seidel morphism of Cartesian products

Rémi Leclercq

Full-text: Open access


We prove that the Seidel morphism of (M×M,ωω) is naturally related to the Seidel morphisms of (M,ω) and (M,ω), when these manifolds are monotone. We deduce that any homotopy class of loops of Hamiltonian diffeomorphisms of one component, with nontrivial image via Seidel’s morphism, leads to an injection of the fundamental group of the group of Hamiltonian diffeomorphisms of the other component into the fundamental group of the group of Hamiltonian diffeomorphisms of the product. This result was inspired by and extends results obtained by Pedroza [Int. Math. Res. Not. (2008) Art. ID rnn049].

Article information

Algebr. Geom. Topol., Volume 9, Number 4 (2009), 1951-1969.

Received: 1 July 2009
Accepted: 31 August 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R17: Symplectic and contact topology
Secondary: 57R58: Floer homology 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms

symplectic manifolds Hamiltonian diffeomorphisms Seidelś morphism


Leclercq, Rémi. The Seidel morphism of Cartesian products. Algebr. Geom. Topol. 9 (2009), no. 4, 1951--1969. doi:10.2140/agt.2009.9.1951.

Export citation


  • P Albers, A Lagrangian Piunikhin–Salamon–Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. (2007) Art. ID rnm134, 56pp
  • P Biran, L Polterovich, D Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J. 119 (2003) 65–118
  • C C Conley, E Zehnder, The Birkhoff–Lewis fixed point theorem and a conjecture of V I Arnol$'$d, Invent. Math. 73 (1983) 33–49
  • M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
  • S Hu, F Lalonde, Anti-symplectic involution and Maslov indices, to appear in Trans. Amer. Math. Soc.
  • F Lalonde, D McDuff, L Polterovich, Topological rigidity of Hamiltonian loops and quantum homology, Invent. Math. 135 (1999) 369–385
  • D McDuff, Quantum homology of fibrations over $S\sp 2$, Internat. J. Math. 11 (2000) 665–721
  • A Pedroza, Seidel's representation on the Hamiltonian group of a Cartesian product, Int. Math. Res. Not. (2008) Art. ID rnn049, 19pp
  • S Piunikhin, D Salamon, M Schwarz, Symplectic Floer–Donaldson theory and quantum cohomology, from: “Contact and symplectic geometry (Cambridge, 1994)”, (C B Thomas, editor), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 171–200
  • D Salamon, Lectures on Floer homology, from: “Symplectic geometry and topology (Park City, UT, 1997)”, (Y Eliashberg, L Traynor, editors), IAS/Park City Math. Ser. 7, Amer. Math. Soc. (1999) 143–229
  • P Seidel, $\pi\sb 1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046–1095
  • E Witten, Topological sigma models, Comm. Math. Phys. 118 (1988) 411–449