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A Combinatorial Introduction to Topology

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Topology is remarkable for its contributions to the popular culture of mathematics. Euler's formula for polyhedra, the four color theorem, the Mobius strip, the Klein bottle, and the general notion of a rubber sheet geometry are all part of the folklore of current mathematics. The student in a first course in topology, however, must often wonder where all the Klein bottles went, for such courses are most often devoted to point set topology, the branch of topology that lies at the foundation of modern analysis but whose intersection with the popular notions of topology is almost empty. In contrast, the present work offers an introduction to combinatorial or algebraic topology, the other great branch of the subject and the source of most of its popular aspects.

There are many good reasons for putting combinatorial topology on an equal footing with point set topology. One is the strong intuitive geometric appeal of combinatorial topology. Another is its wealth of applications, many of which result from connections with the theory of differential equations. Still another reason is its connection with abstract algebra via the theory of groups. Combinatorial topology is uniquely the subject where students of mathematics below graduate level can see the three major divisions of mathematics-analysis, geometry, and algebraworking together amicably on important problems.
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