Abstract
This chapter iscusses the proper homotopy theory. The Proper homotopy theory is both an old and a new area of algebraic topology. Its origins go back to the classification of noncompact surfaces by Kerekjarto in 1923, but it is probably fair to say that it got off the ground as a distinct area of algebraic topology as a result of the geometric work of Larry Siebenmann in 1965. The methods and perspectives of the shape theory interacted with those of proper homotopy theory for the mutual benefit of both. This led, in 1976, to the publication by Edwards and Hastings of their lecture notes, which laid down a theoretical framework for studying the proper homotopy theory that is still actively used today. Furthermore, the other approach to the proper homotopy theory currently being developed is based on the algebraic homotopy theory of Baues, and Baues himself has collected material from earlier sources, together with a wealth of new material, in a draft manuscript. The main point of this approach is the use of the language and results of the theory of cofibration categories.
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