Algebraic Topology - Revisions and Additions
Additional Exercises: Approximately
100 new exercises, along with enhanced versions of a few of the original exercises.
In addition to keeping the online version of the book up to date on corrections as they come to light, I have also included a number of revisions. Most of these are small enough not to require renumbering of pages or theorems. A list of the more significant changes is given below. The 2015 reprinting of the book includes all corrections and revisions of the book up to that date except the final two sections added to the Appendix -- see the last item on the list below. (The publisher did not want to include these since this would have made the book six pages longer, and lengthening a book without having a new edition is apparently not done.)
- Chapter 1: Rephrased the proof of Theorem 1.7 to be more explicitly in terms of covering spaces. Gave a different proof of Proposition 1.14 via a lemma that will be used in van Kampen's theorem in the next section. Expanded Proposition 1.26 to cover attachment of higher dimensional cells as well as 2-cells.
- Chapter 2: Improved the definition of Delta complexes.
- Chapter 3: Rearranged the material in section 3.2 on cup products into a more efficient order, renumbering items 3.11-3.21. Expanded the proof of Proposition 3.22 to include the case n odd as well as n even. Simplified the statement and proof of Lemma 3.27. Revised 3.44-3.46.
- Chapter 4: Gave a cleaner statement and proof of Lemma 4.10. Rearranged the material on CW approximations on pp. 352-355 to improve the exposition, renumbering items 4.13-4.17.
- Appendix: Added Proposition A.17 relating product and quotient spaces,
as well as an example of Dowker about products of CW complexes.
- Appendix: Added a one-page section on the homotopy extension property, generalizing a fact proved in Chapter 0. Added a five-page section on special classes of CW
complexes that are provided with various sorts of simplicial structures. In
particular, simplicial sets are introduced and related to the CW complexes
which are their geometric realizations. (My thanks to Greg Kuperberg for telling
me about this geometric view of simplicial sets.)
The renumbering scheme for items 3.11-3.21 and 4.13-4.17 is given in detail at the end of the Preface of the book in both the online version and the 2015 reprinting.
Not included in either the online or printed versions are the following items which I have at various times thought about including in a revision but for one reason or another haven't actually done:
- A house with one room: A simpler
version of the house with two rooms described on page 4. It happens to be
homeomorphic to a space commonly called the dunce hat.
- Section 1.3: An alternative arrangement
of the material on covering spaces, emphasizing group actions more.
- Cap Products: A version
of the exposition on pages 239-242 that emphasizes the connection with cup
products. There are also smaller changes on pages 248-250. The pdf file includes
the pages in between as well, but these are unchanged.
- Solution to Exercise 1 in Section 3C
on the equivalence between different definitions of an H-space. The solution
is somewhat tricky, so perhaps it should have been incorporated into the text
proper, although I'm not sure the result is important enough to merit this.
- A more general relative Künneth formula. The Künneth formula for the homology of the product of two CW complexes is given in Section 3B of the book, including a relative version for products of CW pairs in the paragraph following Corollary 3B.7. As noted on page 357, the technique of CW approximations extends the absolute version of the Künneth formula to products of arbitrary spaces. However, nothing was said about a relative version for non-CW pairs. This omission is addressed here, where a relative version is proved under an additional topological hypothesis on the pairs and a counterexample is given showing that some such extra condition is necessary. (I would insert this on page 357 of the online version of the book if there were some way to do this without renumbering the pages.)