#### Volume 11, issue 2 (2011)

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Symplectic manifolds with vanishing action–Maslov homomorphism

### Mark Branson

Algebraic & Geometric Topology 11 (2011) 1077–1096
##### Abstract

The action–Maslov homomorphism $I:{\pi }_{1}\left(Ham\left(X,\omega \right)\right)\to ℝ$ 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 $\mathsc{D}$ (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
##### 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