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Abstract

Homology is a very powerful tool in that it allows one to draw conclusions about global properties of spaces and maps from local computations. It also involves a wonderful mixture of algebra, combinatorics, computation, and topology. Each of these subjects is, of course, interesting in its own right and appears as the subject of multiple sections in this book. But our primary objective is to see how they can be combined to produce homology, how homology can be computed efficiently, and how homology provides us with information about the geometry and topology of nonlinear objects and functions. Given the amount of theory that needs to be developed, it is easy to lose sight of these objectives along the way.