1. CJM 2012 (vol 65 pp. 553)
|Addendum and Erratum to "The Fundamental Group of $S^1$-manifolds"|
This paper provides an addendum and erratum to L. Godinho and M. E. Sousa-Dias, "The Fundamental Group of $S^1$-manifolds". Canad. J. Math. 62(2010), no. 5, 1082--1098.
Keywords:symplectic reduction; fundamental group
Categories:53D19, 37J10, 55Q05
2. CJM 2010 (vol 62 pp. 1082)
|The Fundamental Group of $S^1$-manifolds|
We address the problem of computing the fundamental group of a symplectic $S^1$-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian $S^1$-action. Several examples are presented to illustrate our main results.
Categories:53D20, 37J10, 55Q05
3. CJM 2004 (vol 56 pp. 553)
|Cohomology Ring of Symplectic Quotients by Circle Actions |
In this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is isolated; however the circle action only needs to be ``formally Hamiltonian''. By identifying the kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the results of Kalkman and Hausmann-Knutson are special cases of our result.
Categories:53D20, 53D30, 37J10, 37J15, 53D05