Table of Contents
ISRN Geometry
Volume 2011 (2011), Article ID 642834, 24 pages
http://dx.doi.org/10.5402/2011/642834
Research Article

Characteristic Number Associated to Mass Linear Pairs

Departamento de FΓ­sica, Universidad de Oviedo, Avda Calvo Sotelo, 33007 Oviedo, Spain

Received 12 July 2011; Accepted 4 August 2011

Academic Editor: A. Fino

Copyright Β© 2011 AndrΓ©s ViΓ±a. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Ξ” be a Delzant polytope in ℝ𝑛 and π›βˆˆβ„€π‘›. Let 𝐸 denote the symplectic fibration over 𝑆2 determined by the pair (Ξ”,𝐛). Under certain hypotheses, we prove the equivalence between the fact that (Ξ”,𝐛) is a mass linear pair (McDuff and Tolman, 2010) and the vanishing of a characteristic number of 𝐸. Denoting by Ham(𝑀Δ), the Hamiltonian group of the symplectic manifold defined by Ξ”, we determine loops in Ham(𝑀Δ) that define infinite cyclic subgroups in πœ‹1(Ham(𝑀Δ)) when Ξ” satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch sur-face, (ii) it is a Δ𝑝 bundle over Ξ”1, and (iii) Ξ” is the truncated simplex associated with the one point blowup of ℂ𝑃𝑛.