Homotopy Theory and Models

This text is based on notes taken by the author at the DMV Seminar on Algebraic Models of Homotopy Types. The work aims to provide an overview of homotopy theory from the point of view of algebraic models of homotopy types, leading the reader from basic definitions in algebraic topology to specific fields of research. The exposition is addressed towards graduate students as well as to researchers from other fields. Due to the scope and size of the book, only a few complete proofs can be given; however, the fundamental concepts and methods of the subject are pointed out, and a number of recent results are included together with the essential bibliographic references. The book begins with a recollection of the basic notions of homotopy theory. A discussion of rational homotopy theory, emphasizing both the Sullivan and the Quillen models, follows. Next, the Adams-Hilton and the Anick models are shown to yield some integral information. The last chapter provides an introduction to the integral classification of homotopy types, along the lines initiated by J.H.C. Whitehead and considerably developed by H. Baues during the last decade.