Examples of circle actions on symplectic spaces

Abstract

Many interesting results in the study of symplectic torus actions can be proved by purely cohomological methods. All one needs is a closed orientable topological 2nmanifold M (or, more generally, a reasonably pleasant topological space whose rational cohomology satisfies Poincare duality with formal dimension 2n), which is cohomologically symplectic (c-symplectic) in the sense that there is a class w ∈ H(M ;Q) such that w 6= 0. Sometimes one requires that M satisifes the Lefschetz condition that multiplication by wn−1 is an isomorphism H(M ;Q) → H2n−1(M ;Q). And an action of a torus T on M is said to be cohomologically Hamiltonian (c-Hamiltonian) if w ∈ Im[i∗ : H(MT ;Q) → H∗(M ;Q)], where MT is the Borel construction; and i : M →MT is the inclusion of the fibre in the fibre bundle MT → BT . Some examples of some results which can be proved easily by cohomological methods are the following.