The circular restricted three-body problem has five relative equilibria L 1 , L 2 , ..., L 5. The invariant stable-unstable manifolds of the center manifolds originating at the partially hyperbolic equilibria L 1 ,L 2 have been identified as the separatrices for the motions which transit between the regions of the phase–space which are internal or external with respect to the two massive bodies. This paper is devoted to the global computation of these manifolds in the spatial case with a suitable finite time chaos indicator. The definition of the chaos indicator is not trivial, since the mandatory use of the regu-larizing Kustaanheimo-Stiefel variables introduces branch discontinuities in the individual Lyapunov indicators. From the study of such discontinuities, we define geometric chaos indicators which are smooth, globally defined, and whose ridges sharply approximate the stable and unstable manifolds of the center manifolds of L 1 , L 2. We illustrate the method for the Sun-Jupiter mass ratio, and represent the topology of the asymptotic manifolds using sections and three-dimensional representations.