Lecture notes in algebraic topology TeX

This is a textbook for a second-year graduate course in topology, highlighting the basic tools of homotopy theory. The authors remark that, at their home institution, most topology research students are in geometric topology, and the book does emphasize the geometric origins of homotopy theory, as opposed to a more abstract, categorical approach. Prerequisites are the fundamental group, covering spaces (especially covering transformations), CW-complexes, singular homology and cohomology, and some elementary notions from differential topology, such as transversality. <P> The first third of the book is occupied with background material, including universal coefficient theorems, products in singular (co)homology, fiber bundles, and homology with local coefficients. The core material then begins with a chapter on fibrations, cofibrations, homotopy groups, and the basic exact sequences in which these objects are involved. This is followed by a chapter on obstruction theory, leading to Eilenberg-Mac Lane spaces and their role in representing ordinary cohomology. Next come generalized (co)homology theories, beginning with framed bordism, then stable homotopy, then spectra in general. <P> Perhaps the most important material is contained in the two chapters on spectral sequences, or, really, on one particular spectral sequence, namely, a combination of the Leray-Serre and Atiyah-Hirzebruch sequences, which, given a fibration $F \to E \to B$, converges to the generalized (co)homology $h(E)$. The focus here is on understanding how to use spectral sequences rather than on setting up the theory behind how they arise. The point is that the consequences of spectral sequences are abundant and wide-ranging. The topics presented to support this include the Hurewicz theorem, the Freudenthal suspension theorem, some calculations of homotopy groups of spheres, the Steenrod squares, the Thom isomorphism theorem, Stiefel-Whitney classes, and the relation between intersection numbers and cup products. The book concludes on a somewhat different note with a chapter on simple-homotopy theory, reaching Reidemeister torsion and the classification of lens spaces, first up to homotopy, then up to homeomorphism. <P> The book is written in a down-to-earth, non-formal style, and the material has been kept to a manageable length. In a handful of places, full proofs of complicated results are omitted, and the reader is referred to the literature, for instance, G. W. Whitehead's book \ref[<em> Elements of homotopy theory</em>, Springer, New York, 1978; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=516508&loc=fromrevtext">MR0516508 (80b:55001)</A>]. The most conspicuous example of this is the construction of the Leray-Serre-Atiyah-Hirzebruch spectral sequence mentioned above. Roughly 200 exercises are included; they are well integrated into the exposition. In addition, each chapter concludes with one or two "projects," in which the reader is asked to look up and digest a substantial result and which could serve as sources for student presentations. Some of these involve material which is used later in the text (limits and colimits, classifying spaces for fiber bundles), and some are included more for breadth (Postnikov systems, the $EHP$ sequence). <P> The authors have made a special effort to keep the presentation flexible, holding down the amount of technical terminology and notation "to minimize the need to shuffle pages back and forth when reading the book". This feature is likely to be appealing not only to students, but also to practicing mathematicians in need of a refresher on a particular topic.