π1 of Hamiltonian S 1 manifolds

Let (M,ω) be a connected, compact symplectic manifold equipped with a Hamiltonian S action. We prove that, as fundamental groups of topological spaces, π1(M) = π1(minimum) = π1(maximum) = π1(Mred), where Mred is the symplectic quotient at any value in the image of the moment map φ. Let (M,ω) be a connected, compact symplectic manifold equipped with a circle action. If the action is Hamiltonian, then the moment map φ : M → R is a perfect Bott-Morse function. Its critical sets are precisely the fixed point sets M 1 of the S action, and M 1 is a disjoint union of symplectic submanifolds. Each fixed point set has even index. By [1], φ has a unique local minimum and a unique local maximum. We will use Morse theory to prove Theorem 0.1. Let (M, ω) be a connected, compact symplectic manifold equipped with a Hamiltonian S action. Then, as fundamental groups of topological spaces, π1(M) = π1(minimum) = π1(maximum) = π1(Mred), where Mred is the symplectic quotient at any value in the image of the moment map φ. Remark 0.2. The theorem is not true for orbifold π1 of Mred, as shown in the example below. (See [5] or [11] for the definition of orbifold π1). Let a ∈ im(φ), and φ(a) = {x ∈ M | φ(x) = a} be the level set. Define Ma = φ (a)/S to be the symplectic quotient. Note that if a is a regular value of φ, and if the circle action on φ(a) is not free, then Ma is an orbifold, and we have an orbi-bundle: (0.1) S →֒ φ(a) ↓ Ma If a is a critical value of φ, then Ma is a stratified space. ([10]). Now, let S act on (S × S, 2ρ ⊕ ρ) (where ρ is the standard symplectic form on S) by λ(z1, z2) = (λ z1, λz2). Let 0 be the minimal value of the moment map. Then for a ∈ (1, 2), Ma is an orbifold which is homeomorphic to S 2 and has two Z2 singularities. The orbifold π1 of Ma is Z2, but the π1 of Ma as a topological space is trivial. Let a be a regular or a critical value of φ. Define M = {x ∈ M | φ(x) ≤ a}. By Morse theory, we have the following lemmas about how M and φ(a) change when φ doesn’t cross or crosses a critical level. Date: January 10, 2002 and, in revised form, May 16, 2002. 1991 Mathematics Subject Classification. Primary : 53D05, 53D20; Secondary : 55Q05, 57R19.