The fundamental group of sympletic manifolds with Hamiltonian lie group actions

Abstract

Let (M, ω) be a connected, compact symplectic manifold equipped with a Hamiltonian G action, where G is a connected compact Lie group. Let φ be the moment map. In [12], we proved the following result for G = S 1 action: as fundamental groups of topological spaces, π 1 (M) ∼ = π 1 (M red ), where M red is the symplectic quotient at any value of the moment map φ, and ∼ = denotes "isomorphic to". In this paper, we generalize this result to other connected compact Lie group G actions. We also prove that the above fundamental group is isomorphic to that of M/G. We briefly discuss the generalization of the first part of the results to non-compact manifolds with proper moment maps.

The Fundamental Group of G-Manifolds

“…For a compact Hamiltonian G-manifold M , we proved the following results, which combine Theorem 0.1 in [13] and Theorems 1.2, 1.3 and 1.6 in [14]. Theorem 1.1.…”

“…We can similarly prove the claims for N = φ −1 (U ). See the proof of Lemma 3.8 in [14] for the computation of L H in this case and argue similarly as above.…”

“…Given a value a in g * , the space M a = φ −1 (G · a)/G is called the symplectic quotient or the reduced space at the coadjoint orbit G · a. If G a is the stabilizer group of a under the coadjoint action, then we also have that M a = φ −1 (a)/G a .In this paper, unless otherwise stated, G always denotes a connected compact Lie group, im(φ) means the image of φ, and fundamental group always means the fundamental group of a space as a topological space.For a compact Hamiltonian G-manifold M , we proved the following results, which combine Theorem 0.1 in [13] and Theorems 1.2, 1.3 and 1.6 in [14]. Theorem 1.1.…”

“…Using the Van-Kampen theorem, we can prove the following lemma. May see Lemma 2.3 in [14] for a proof.…”

“…In this case, im(φ) is compact, its intersection with a closed positive Weyl chamber is a convex polytope (Theorem 2.1). Let b be a boundary vertex furthest from the origin on the polytope, then φ −1 (b) is a fixed set of G b , where G b is the stabilizer group of b under the coadjoint action (see Lemma 6.14 in [14]. Note that G b contains a maximal torus of G).…”