My goal in this book is to enable a non-specialist to grasp and participate in current research in computational topology. Therefore, this book is not a compilation of recent advances in the area. Rather, the book presents basic mathematical concepts from a computer scientist's point of view, focusing on computational challenges and introducing algorithms and data structures when appropriate. The book also incorporates several recent results from my doctoral dissertation and subsequent related results in computational topology.

The primary motivation for this book is the significance and utility of topological concepts in solving problems in computer science. These problems arise naturally in computational geometry, graphics, robotics, structural biology, and chemistry. Often, the questions themselves have been known and considered by topologists. Unfortunately, there are many barriers to interaction:

• Computer scientists do not know the language of topologists. Topology, unlike geometry, is not a required subject in high school mathematics and is almost never dealt with in undergraduate computer science. The axiomatic nature of topology further compounds the problem as it generates cryptic and esoteric terminology that makes the field unintelligible and inaccessible to non-topologists.

• Topology can be very unintuitive and enigmatic and therefore can appear very complicated and mystifying, often frightening away interested computer scientists.

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