Symplectic manifolds with vanishing action–Maslov homomorphism

Branson, Mark

doi:10.2140/agt.2011.11.1077

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Symplectic manifolds with vanishing action–Maslov homomorphism

Mark Branson

Algebraic & Geometric Topology 11 (2011) 1077–1096

DOI: 10.2140/agt.2011.11.1077

Abstract

The action–Maslov homomorphism $I : π_{1} (Ham (X, ω)) \to ℝ$ is an important tool for understanding the topology of the Hamiltonian group of monotone symplectic manifolds. We explore conditions for the vanishing of this homomorphism, and show that it is identically zero when the Seidel element has finite order and the homology satisfies property $D$ (a generalization of having homology generated by divisor classes). We use these results to show that $I = 0$ for products of projective spaces and the Grassmannian of $2$ planes in $ℂ^{4}$ .

Keywords

action–Maslov, quantum homology, floer theory, Seidel homomorphism, symplectic geometry

Mathematical Subject Classification 2000

Primary: 53D45

Secondary: 53D35, 53D40, 20F69

References

Publication

Received: 2 November 2010

Revised: 31 January 2011

Accepted: 3 February 2011

Published: 30 March 2011

Authors

Mark Branson
Department of Mathematics The Technion, Israeli Institute of Technology Haifa, 32000 Israel

Volume 11, issue 2 (2011)