ABSTRACT
This chapter focuses on the fundamental concepts and computational algorithms in surface algebraic topology. Algebraic topology studies the topology of a space using algebraic methods. The basic philosophy for algebraic topology is to associate algebraic groups with topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. In medical imaging field, the human organ surfaces can be extracted from volumetric CT or MRI images. First, the images are segmented into foreground and background; second, the boundary surface of the foreground volume is extracted. The middle surface is more complicated, its boundary has 3 connected components. In fact, it has one handle. For two orientable surfaces with boundaries, they are topologically equivalent, if and only if they have the same genus and the same number of boundary connected components. A surface with only one side is called non-orientable. The Möbius band is one of the most common non-orientable surfaces.
