Celebratio Mathematica — McDuff

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D. McDuff: “A survey of the topological properties of symplectomorphism groups,” pp. 173–193 in Topology, geometry and quantum field theory: Proceedings of the 2002 Oxford symposium in honour of the 60th birthday of Graeme Segal (Oxford, 24–29 June 2002). Edited by U. Tillmann. London Mathematical Society Lecture Note Series 308. Cambridge University Press, 2004. MR 2079375 Zbl 1102.57013 ArXiv SG/0404340 incollection

The special structures that arise in symplectic topology (particular Gromov–Witten invariants and quantum homology) place as yet rather poorly understood restrictions on the topological properties of symplectomorphism groups. This article surveys some recent work by Abreu, Lalonde, McDuff, Polterovich and Seidel, concentrating particularly on the homotopy properties of the action of the group of Hamiltonian symplectomorphisms on the underlying manifold \( M \). It sketches the proof that the evaluation map \[ \pi_1(\mathrm{Ham}(M))\to\pi_1(M) \] given by \[ \{\phi_t\}\mapsto\{\phi_t(x_0)\} \] is trivial, as well as explaining similar vanishing results for the action of the homology of \( \mathrm{Ham}(M) \) on the homology of \( M \). Applications to Hamiltonian stability are discussed.

Ulrike Tillmann	Related
Graeme Bryce Segal	Related