No Access

THE FUNDAMENTAL GROUP OF G-MANIFOLDS

HUI LI

School of Mathematics, Suzhou University, Box 173, Suzhou 215006, P. R. China

https://doi.org/10.1142/S0219199712500563Cited by:2 (Source: Crossref)

Abstract

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 for all a ∈ ϕ(M), where 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).

Keywords:

AMSC: 53D05, 53D20, 55Q05, 57R19

References

M. A. Armstrong, Proc. Amer. Math. Soc. 84(2), 267 (1982). Crossref, Web of Science, Google Scholar
M. Atiyah , Bull. London Math. Soc. 14 , 1 ( 1982 ) . Crossref, Web of Science, Google Scholar
T. Bröcker and T. Tom Dieck , Representations of Compact Lie Groups , Graduate Texts in Mathematics 98 ( Springer , 1985 ) . Crossref, Google Scholar
G. Bredon , Introduction to Compact Transformation Groups ( Academic Press , New York , 1972 ) . Google Scholar
V. Guillemin , E. Lerman and S. Sternberg , Symplectic Fibrations and Multiplicity Diagrams ( Cambridge University Press , Cambridge , 1996 ) . Crossref, Google Scholar
V. Guillemin and S. Sternberg, Invent. Math. 67(3), 491 (1982). Crossref, Web of Science, Google Scholar
V. Guillemin and S. Sternberg, Differential Geometric Methods in Mathematical Physics, ed. S. Sternberg (Reidel, Dordrecht, 1984) pp. 161–175. Crossref, Google Scholar
V. Guillemin and S. Sternberg , Symplectic Techniques in Physics ( Cambridge University Press , Cambridge , 1990 ) . Google Scholar
L. Godinho and M. E. Sousa-Dias , Canad. J. Math. 62 , 1082 ( 2010 ) . Crossref, Web of Science, Google Scholar
F. C. Kirwan , Invent. Math. 77 , 547 ( 1984 ) . Crossref, Web of Science, Google Scholar
E. Lerman, Enseign. Math. 51(2), 117 (2005). Google Scholar
E. Lermanet al., Topology 37(2), 245 (1998). Crossref, Google Scholar
H. Li, Proc. Amer. Math. Soc. 131(11), 3579 (2003). Crossref, Web of Science, Google Scholar
H. Li, J. Symplectic Geom. 4(3), 345 (2007). Crossref, Google Scholar
C. M. Marle, Séminaire sud-rhodanien de géométrie, Π, Travaux en cours (Hermann, Paris, 1984) pp. 19–35. Google Scholar
R. Sjamaar and E. Lerman, Ann. of Math. 134(2), 375 (1991). Crossref, Web of Science, Google Scholar
C. Woodward, Amer. J. Math. 128(2), 311 (2006). Crossref, Web of Science, Google Scholar