Celebratio Mathematica — McDuff

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D. McDuff: “Loops in the Hamiltonian group: A survey,” pp. 127–148 in Symplectic topology and measure preserving dynamical systems (Snowbird, UT, 1–5 July 2007). Edited by A. Fathi, Y.-G. Oh, and C. Viterbo. Contemporary Mathematics 512. American Mathematical Society (Providence, RI), 2010. MR 2605315 Zbl 1202.53083 ArXiv 0711.4086 incollection

This note describes some recent results about the homotopy properties of Hamiltonian loops in various manifolds, including toric manifolds and one point blow ups. We describe conditions under which a circle action does not contract in the Hamiltonian group, and construct an example of a loop \( \gamma \) of diffeomorphisms of a symplectic manifold \( M \) with the property that none of the loops smoothly isotopic to \( \gamma \) preserve any symplectic form on \( M \). We also discuss some new conditions under which the Hamiltonian group has infinite Hofer diameter. Some of the methods used are classical (Weinstein’s action homomorphism and volume calculations), while others use quantum methods (the Seidel representation and spectral invariants).

Yong-Geun Oh	Related
Claude Viterbo	Related
Albert Fathi	Related