Corrections to the later printings of
*Algebraic Topology*

Last updated April 20, 2018.

Below is a list of corrections, clarifications, and comments for the printings of the book starting in late 2015. Undoubtedly more errors remain to be discovered. If you find any, please send them to me for inclusion in this list and for correcting future printings and the online version of the book.

- Table of Contents. In Chapter 1 the item "Applications to Cell Complexes" is on page 49 rather than 50, as of late 2015.
- Chapter 0, page 16. For further explanation of the point-set topology underlying the second sentence of the proof of Proposition 0.18, see the pdf file of cumulative corrections for the book. (8/31/2017)
- Section 2.1, page 109, line 5. The phrase "exactly two" is not quite correct since the two faces in a canceling pair could be faces of the same simplex. To fix this, replace this sentence by the following sentence: "If $ \x $ is a cycle, all the (n-1)-dimensional faces of the Delta^n_i 's are identified in pairs." The online version of the book also contains some slight rewordings in the remainder of this paragraph, for the sake of clarity. (3/11/2016)
- Section 2.1, page 125, Example 2.23. In the first paragraph of this example the sentence beginning "The second isomorphism" needs to be modified in the special case n=1 since the boundary of D^{n-1} is empty in this case, which means that (D^{n-1},boundary D^{n-1}) is not a good pair when n=1. However the claimed isomorphism is easy to see in this case since it involves just H_0. The online version of the book has been rephrased to deal with this issue. (The old version of this paragraph has 7 lines after the displayed formulas, the revised version has 8 lines.) (3/11/2016)
- Section 2.2, page 137. Part (c) of Lemma 2.34 has been expanded to say also that the inclusion-induced map on homology is surjective when k=n. The proof on the next page has been revised to show this.
- Section 2.2, page 144, Example 2.42. For clarity it should say that q\phi restricts to a homeomorphism from each component of S^{k-1}-S^{k-2} onto RP^{k-1}-RP^{k-2}. (4/20/2018)
- Section 2.C, page 180, line -11. Typo: The formula involving tau should be just tau(fr) = tau(f), without the star subscripts. (3/4/2017)
- Section 4.1, page 348. The first paragraph of the subsection on Cellular Approximation has been revised to eliminate a reference to an earlier proof of Proposition 1.14 that has been replaced by a different proof in later versions of the book. (1/7/18)
- Section 4.2, page 385. In versions of the book before 2016 the chart showing the 2-primary parts of the stable homotopy groups of spheres had a couple of errors in the range above dimension 50. The original calculations in this range were done by Kochman and Mahowald in the 1990's. When these groups were recalculated by Dan Isaksen by different methods in a 2014 arXiv preprint called "Stable stems", a few discrepancies were found. Isaksen's calculations have been checked by other experts, so there is a high probability that they are correct. A corrected version of the chart now appears in the online version of the book. A few changes were also made in the accompanying text in pages 385-388. (3/11/2016)