xx Preface spaces is established and the concept of homotopy is developed should be covered fairly thoroughly. Chapter 5 is on cofibrations and fibrations. The basic properties should be explored, and the mapping cylinder and its dual should be studied carefully it’s probably best to gloss over the distinction between the pointed and unpointed cases. State the Fundamental Lift- ing Property and the basic factorization theorems without belaboring their proofs. The fact that fiber and cofiber sequences lead to exact sequences of homotopy sets should be explored in detail. Chapter 6 is on homotopy colimits and limits. Cover homotopy pushouts in detail, appealing to du- ality for homotopy pullbacks, and give a brief discussion of the issues for more general diagrams. Chapter 7 is on homotopy pullback and pushout squares and should be covered in some detail. Chapters 8 and 9 offer a huge collection of topics. For the moment, only Section 8.1 (Long cofiber and fiber sequences) and perhaps Section 9.2 (on H-Spaces and co-H-spaces) are really mandatory. Other sections can be covered as needed or assigned to students as homework. Chapter 10 is a brief account of abstract model cat- egories. It is included for ‘cultural completeness’ and, since it does not enter into the main flow of the text, it can be skipped in its entirety. Part 3 covers the four major special features of the homotopy theory of spaces. Chapters 11 through 14 should be covered in detail. Chapter 15 is a combination of topics and cultural knowledge. Sections 15.1 and 15.2 are crucial, but the rest can be glossed over if need be. Part 4 is where the four basic topological inputs are developed into effective tools for studying homotopy-theoretical problems. Chapters 16 through 19 should all be covered in detail. Chapter 20 contains topics which can be assigned to students as homework. The second semester should pick up with Part 5 where we develop coho- mology (and homology). Chapters 21 through 24 should be covered pretty thoroughly. Chapter 25 is a vast collection of topics, which can be covered at the instructor’s discretion or assigned as homework. Part 6 is about the cohomology of fiber sequences, leading ultimately to the Leray-Serre spec- tral sequence, which is notoriously forbidding when first encountered. The exposition here is broken into small pieces with a consistent emphasis on the topological content. Many of the basic ideas and a nice application are covered in Chapters 26 through 29 this would be a fine place to stop if time runs out. Otherwise, Chapters 30 and 31 get to the full power of the Leray-Serre spectral sequence. This power is used in Chapter 32 to prove the Bott Periodicity Theorem. Chapter 33 is another topics chapter, which includes the cohomology of Eilenberg-MacLane spaces and some computa- tions involving the homotopy groups of spheres. Finally, Part 7 covers some very fun and interesting topics: localization and completion, a discussion of the exponents of homotopy groups of spheres including a proof of Selick’s theorem on the exponent of π∗(S3) the theory of closed classes and a dual

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