The purpose of this paper is to solve the global asymptotic stabilization problem for nonlinear systems on general manifolds. It is known that if the state space of a control system is not contractible, the system is not globally asymptotically stabilizable via C¹ feedback law, because gradient-like flow on the non-contractible manifold demands multiple singular points. In this paper, we define a control Lyapunov-Morse function having multiple critical points using the concept of the Lyapunov-Morse function, which is a kind of complete Lyapunov functions for dynamical systems with multiple isolated singular points. We derive a discontinuous feedback law from the control Lyapunov-Morse function. Moreover, a condition for global asymptotic stability of the controlled system with the discontinuous feedback law is also obtained.