THE FUNDAMENTAL GROUP OF G-MANIFOLDS
2013 ◽
Vol 15
(03)
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pp. 1250056
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Keyword(s):
Let G be a connected compact Lie group, and let M be a connected Hamiltonian G-manifold with equivariant moment map ϕ. We prove that if there is a simply connected orbit G ⋅ x, then π1(M) ≅ π1(M/G); if additionally ϕ is proper, then π1(M) ≅ π1 (ϕ-1(G⋅a)), where a = ϕ(x). We also prove that if a maximal torus of G has a fixed point x, then π1(M) ≅ π1(M/K), where K is any connected subgroup of G; if additionally ϕ is proper, then π1(M) ≅ π1(ϕ-1(G⋅a)) ≅ π1(ϕ-1(a)), where a = ϕ(x). Furthermore, we prove that if ϕ is proper, then [Formula: see text] for all a ∈ ϕ(M), where [Formula: see text] is any connected subgroup of G which contains the identity component of each stabilizer group; in particular, π1(M/G) ≅ π1(ϕ-1(G⋅a)/G) for all a ∈ ϕ(M).
2020 ◽
pp. 205-212
Keyword(s):
2018 ◽
Vol 2018
(742)
◽
pp. 157-186
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Keyword(s):
1992 ◽
Vol 34
(3)
◽
pp. 379-394
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Keyword(s):
2002 ◽
Vol 13
(03)
◽
pp. 217-225
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Keyword(s):
2000 ◽
Vol 02
(01)
◽
pp. 75-86
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2011 ◽
Vol 148
(3)
◽
pp. 807-834
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Keyword(s):