Business. Symplectic Geometry Rational Cohomology

In this paper, we mainly discuss the procedure for computing the rational cohomology of quotients group actions in symplectic geometry. This approach isthrough considering the function f(x) = ||μ(x)||2as a Morse function and apply Morse theory to it. But / has singularities , i.e. , / is not nondegenerate, so the results ofMorse theory cannot be applied to it directly. But we can see from this paper that / isa minimally degenerate function, and that Morse theory can be extended to cover this class of functions. If the symplectic quotient for the action of a compact Lie group on asymplectic manifold exists, let the moment map of this action be μ, then the rational equivariant cohomology of μ-1 (0) is isomorphic to the ordinary rational cohomologyof the symplectic quotient μ-1(0)/K. Thus the computation of the latter can beconverted into the computation of the former, which is obtained just through using the Morse theory of minimally degenerate functions.