Abstract view
Cohomology Ring of Symplectic Quotients by Circle Actions


Published:20040601
Printed: Jun 2004
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 TolmanWeitsman 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 HausmannKnutson 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
