The topology of reduced phase spaces of the motion of vortices on a sphere (Q1107886)

The motion of point vortices in a perfect incompressible fluid on a two- dimensional sphere can be represented by a Hamiltonian flow which is invariant under the action of the special orthogonal group SO(3). Following the reduction method of Marsden and Weinstein this motion can be described by Hamiltonian flows on reduced phase spaces. Some topological invariants of these reduced phase spaces are calculated. For suitable vortices the author believes that the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system. The layout of this paper is as follows: Section 1 describes Bogomolov's model and the reduced phase space. Section 2 recalls the definition of the Betti numbers of a manifold and why they are related to the numbers of fixed points of Hamiltonian flow. Section 3 gives the main results and an outline of their proof. Section 4 gives a general procedure for computing the Betti numbers of reduced phase spaces which is used to prove the assertions in Section 3.