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Surface mapping is fundamental to shape computing and various downstream applications. This paper develops a pants decomposition framework for computing maps between surfaces with arbitrary topologies. The framework first conducts pants decomposition on both surfaces to segment them into consistent sets of pants patches (a pants patch is intuitively defined as a genus-0 surface with three boundaries), then composes global mapping between two surfaces by using harmonic maps of corresponding patches. This framework has several key advantages over existing techniques. First, it is automatic. It can automatically construct mappings for surfaces with complicated topology, guaranteeing the one-to-one continuity. Second, it is general and powerful. It flexibly handles mapping computation between surfaces with different topologies. Third, it is flexible. Despite topology and geometry, it can also integrate semantics requirements from users. Through a simple and intuitive human-computer interaction mechanism, the user can flexibly control the mapping behavior by enforcing point/curve constraints. Compared with traditional user-guided, piecewise surface mapping techniques, our new method is less labor intensive, more intuitive, and requires no user's expertise in computing complicated surface maps between arbitrary shapes. We conduct various experiments to demonstrate its modeling potential and effectiveness.