Integral manifolds of the restricted three-body problem

We compute the homology of the integral manifolds of the restricted three-body problem-planar and spatial, unregularized and regularized. Holding the Jacobi constant fixed defines a three-dimensional algebraic set in the planar case and a five dimensional algebraic set in the spatial case (the integral manifolds). The singularities of the restricted problem due to collusions are removable, which defines the regularized problem. There are five positive critical values of the Jacobi constant: one is due to a critical point at infinity, another is due to the Lagrangian critical points and three are due to the Eulerian critical points. The critical point at infinity occurs only in spatial problems. We compute the homology of the integral manifold for each regular value of the Jacobi constant. These computations show that at each critical value the integral manifolds undergo a bifurcation in their topology. The bifurcation due to a critical point at infinity shows that Birkhoff's conjecture is false even in the restricted problem. Birkhoff also asked if the planar problem is the boundary of a cross section for the spatial problem. Our computations and homological criteria show that this can never happen in the restricted problem, but may be possible in the regularized problem for some values of the Jacobi constant. We also investigate the existence of global cross sections in each of the problems.