Cohomology Rings of Symplectic Cuts

The "symplectic cut" construction [Le] produces two symplectic orbifolds C \Gamma and C+ from a symplectic manifoldM with a Hamiltonian circle action. We compute the rational cohomology ring of C+ in terms of those of M and C \Gamma . 1 Statement of the results Let M be a symplectic n-manifold endowed with an Hamiltonian S 1 -action with moment map f : M ! R. Suppose that 0 is a regular value of f and consider the manifolds-with-boundary M \Gamma := f \Gamma1 (R0 ) ; M+ := f \Gamma1 (R0 ) ; M 0 := f \Gamma1 (0) = M \Gamma " M+ : The symplectic cutting of (M; f) at 0, introduced by E. Lerman [Le], can be described as follows. The S 1 -action restricted to M 0 gives rise to the equivalence relation on M : x ¸ y () 8 ? ! ? : x 2 M 0 and y = ` \Delta x for some ` 2 S 1 or x = y if x = 2 M 0 : As 0 is a regular value of f , the stabilizers of points in M 0 are finite cyclic subgroups of S 1 . Therefore, the following quotients are orbifolds: C \Gamma := M \Gamma / ¸ ; ..