Topological concepts and techniques have been broadly applied in computer graphics and geometric modeling. However, the homotopy type of a mapping between two surfaces has not been addressed before. In this paper, we present a novel solution to the problem of computing continuous maps with different homotopy types between two arbitrary triangle meshes with the same topology. Inspired by the rich theory of topology as well as the existing body of work on surface mapping, our newly-developed mapping techniques are both fundamental and unique, offering many attractive advantages. First, our method allows the user to change the homotopy type or global structure of the mapping with minimal intervention. Moreover, to locally affect shape correspondence, we articulate a new technique that robustly satisfies hard feature constraints, without the use of heuristics to ensure validity. In addition to acting as a useful tool for computer graphics applications, our method can be used as a rigorous and practical mechanism for the visualization of abstract topological concepts such as homotopy type of surface mappings, homology basis, fundamental domain, and universal covering space. At the core of our algorithm is a procedure for computing the canonical homology basis and using it as a common cut graph for any surface with the same topology. We demonstrate our results by applying our algorithm to shape morphing in this paper.