3D objects of the same kind often have different topologies, and finding correspondence between them is important for operations such as morphing, attribute transfer, and shape matching. This paper presents a novel method to find the surface correspondence between topologically different surfaces. The method is characterized by deforming the source polygonal mesh to match the target mesh by using the intermediate implicit surfaces, and by performing a topological surgery at the appropriate locations on the mesh. In particular, we propose a mathematically well-defined way to detect the topology change of surface by finding the non-degenerate saddle points of the velocity fields that tracks implicit surfaces. We show the effectiveness and possible applications of the proposed method through several experiments.