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Cohomology Ring of Symplectic Quotients by Circle Actions

  Published:2004-06-01
 Printed: Jun 2004
  • Ramin Mohammadalikhani
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Abstract

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.
MSC Classifications: 53D20, 53D30, 37J10, 37J15, 53D05 show english descriptions Momentum maps; symplectic reduction
Symplectic structures of moduli spaces
Symplectic mappings, fixed points
Symmetries, invariants, invariant manifolds, momentum maps, reduction [See also 53D20]
Symplectic manifolds, general
53D20 - Momentum maps; symplectic reduction
53D30 - Symplectic structures of moduli spaces
37J10 - Symplectic mappings, fixed points
37J15 - Symmetries, invariants, invariant manifolds, momentum maps, reduction [See also 53D20]
53D05 - Symplectic manifolds, general
 

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