Cobalt is a widely used catalyst for many heterogeneous reactions, including the Fischer-Tropsch (FT) process, which converts syngas (H2 and CO) to higher hydrocarbons. As a result, a better understanding of the key chemical steps on the Co surface, such as the dissociative chemisorption of H2 as an initial step of the FT process, is of fundamental importance. Here, we report an accurate full-dimensional global potential energy surface for the dissociative chemisorption of H2 on the rigid Co(0001) surface constructed from more than 3000 density functional theory points. The high-fidelity potential energy surface was obtained using the permutation invariant polynomial-neural network method, which preserves both the permutation symmetry of H2 and translational symmetry of the Co(0001) surface. The reaction path features a very low barrier on the top site. Full-dimensional quantum dynamical calculations provide insights into the dissociation dynamics and influence of the initial vibrational, rotational, and orientational degrees of freedom.