Transversality theory, cobordisms, and invariants of symplectic quotients

This paper gives methods for understanding invariants of symplectic quotients. The symplectic quotients that we consider 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 [23] examines symplectic quotients by a nonabelian group, showing how to reduce to the maximal torus. Throughout this paper we assume X is a symplectic manifold, and that a compact torus T ∼= S × . . . × S acts on X , preserving the symplectic form, and having moment map μ : X → t, where t denotes the dual of the Lie algebra of T . We assume that μ is a proper map. (For definitions and our sign conventions see the notation section at the end of this introduction). For every regular value p ∈ t of the moment map, the inverse image μ(p) is a compact submanifold of X which is stable under T , and on which the T -action is locally free (that is, every point in μ(p) has finite stabilizer subgroup). The symplectic quotient, which we denote X//T (p), is defined by taking the topological quotient by T