Transversality theory, cobordisms, and invariants of symplectic quotients

Abstract

This paper gives methods for understanding invariants of symplectic quotients. The symplectic quotients considered here are compact symplectic manifolds (or more generally orbifolds), which arise as the symplectic quotients of a symplectic manifold by a compact torus. (A companion paper examines symplectic quotients by a nonabelian group, showing how to reduce to the maximal torus.) Let X be a symplectic manifold, with a Hamiltonian action of a compact torus T. The main topological result of this paper describes an explicit cobordism that exists between a symplectic quotient of X by T, and a collection of iterated projective bundles over components of the set of T-fixed-points. The characteristic classes of these bundles can be determined explicitly, and another result uses this to give formulae for integrals of cohomology classes over the symplectic quotient, in terms of data localized at the T-fixed points of X.