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In this paper we first establish a scheme which can be used to classify much of the work that has been done in fixed point theory and then survey the current status of knowledge about semicomplexes in this context. Some recent discoveries are introduced and used to give strengthened forms of some earlier results. Finally, several open questions are posed, whose solutions would, in the opinion of the author, make valuable contributions to the theory of semicomplexes. 1. Classification of fixed point theories. As any check of the recent literature shows, work in fixed point theory has proliferated to such an extent that the field now encompasses a very broad range of mathematical activity. This diversity suggests the need for some type of framework to use in classifying both new and old results in fixed point theory. As an initial step in this direction, we will introduce two axes which seem to form natural dividing lines for the subject. First note that one may approach fixed point questions by either direct or indirect methods. Work will be classified as using the direct approach if it studies fixed point problems in the natural settings where they arise outside of fixed point theory itself. Thus, for example, a question about fixed points for maps between absolute neighborhood retracts (ANR's) would be studied and solved by working entirely within the category of ANR's and their maps. This is contrasted with the indirect approach, which we define as the study of fixed point questions in categories that were originally devised for the purpose of doing some type of fixed point theory. In this case, a fixed point question about maps of ANR's might be regarded as a special case of a question about maps of, for example, semicomplexes and then studied in the category of semicomplexes and their maps — a category specifically invented for fixed point theory. Received by the editors February 2, 1972. AMS 1970 subject classification. Primary 54-02, 54H25, 55C20; Secondary 47H10.

Fixed point theory via semicomplexes