Differentiable pseudo-free circle actions on homotopy spheres

Liang, Chao Chu

doi:10.1090/S0002-9939-1978-0507339-8

Differentiable pseudo-free circle actions on homotopy spheres
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Proc. Amer. Math. Soc. 72 (1978), 362-364 Request permission

Abstract:

Let G denote the circle group, $\varphi$ a differentiable pseudo-free G-action of type $({p_1}, \ldots ,{p_k})$ on a homotopy sphere ${\Sigma ^{2n + 1}}$, and X the vector field induced by $\varphi$. If w is a G-invariant 1-form satisfying $w(X) = 1$, then we will prove that ${\smallint _\Sigma }w \wedge {(dw)^n} = \pm {({p_1}{p_2} \cdots {p_k})^{ - 1}}$.

References

Walter L. Baily Jr., The decomposition theorem for $V$-manifolds, Amer. J. Math. 78 (1956), 862–888. MR 100103, DOI 10.2307/2372472
W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2) 68 (1958), 721–734. MR 112160, DOI 10.2307/1970165
P. E. Conner and E. E. Floyd, Orbit spaces of circle groups of transformations, Ann. of Math. (2) 67 (1958), 90–98. MR 92922, DOI 10.2307/1969928
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1393940
Deane Montgomery and C. T. Yang, Homotopy equivalence and differentiable pseudo-free circle actions on homotopy spheres, Michigan Math. J. 20 (1973), 145–159. MR 324719
Deane Montgomery and C. T. Yang, Differentiable pseudo-free circle actions on homotopy seven spheres, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, pp. 41–101. MR 0362383
I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363. MR 79769, DOI 10.1073/pnas.42.6.359
Ichirô Satake, The Gauss-Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan 9 (1957), 464–492. MR 95520, DOI 10.2969/jmsj/00940464
Alan Weinstein, Symplectic $V$-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math. 30 (1977), no. 2, 265–271. MR 455019, DOI 10.1002/cpa.3160300207