Corrections to the later printings of
*Algebraic Topology*

Last updated June 2, 2021.

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 1.2, page 43. In Example 1.21 I forgot to mention that the spaces X_\alpha should be assumed to be path-connected. (10/22/2019)
- Section 1.2, page 49, line 3. The reference should be to Corollary 3.46 rather than 3.45. (1/5/2020)
- Section 1.3, page 57. In the discussion of covering spaces of the wedge sum of two circles, instead of saying that four edges meet at each vertex it would be more accurate to say that there are four ends of edges at each vertex since an edge with both its ends at the same vertex should be allowed. The text has been modified at several points on this page to clarify this. (7/12/2019)
- Section 1.3, page 62. Near the top of the page, in the two sentences that originally preceded the proof of Proposition 1.33 I mentioned a slightly more general definition of local path-connectedness in which any two points in V were joinable by a path in U rather than in V. It seems I was mistaken about the terminology and this weaker condition has a different name: 0-local connectedness, or 0-LC for short. Since this more general condition is not used in the book I have deleted these two sentences. (12/17/2018)
- Section 1.3, page 82, Exercise 27. (August 2019) This exercise has been revised in the online version of the book to clarify some confusion about the definition of the first action. This revision affects the answer. (8/30/2019)
- Section 1.3, page 82, Exercise 28. The reference should be to part (c) of Proposition 1.40, not part (b). (9/1/2019)
- 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 xi 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 2.1, page 131, Exercise 2. The identifications should preserve orderings of vertices, though this was already implicit in the goal of getting a Delta-complex structure on the quotient. (4/12/2020)
- Section 2.2, page 136. The proof of Proposition 2.30 has been expanded slightly to give more detail on why the map p_i is projection onto the i-th factor. (11/21/2019)
- 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 3.2, page 224. In the proof of Proposition 3.22 the reference should be to Theorem 3.15 rather than 3.16. (12/26/2019)
- Section 3.3, pages 237-8. Mild revisions in steps (2)-(4) in the proof of Lemma~3.27 for clarity. (1/8/2020)
- Section 3.3, page 256, two lines above the proof of Corollary 3.46. Typo: The semi-colon in H_1(X_{m,n};Z) was missing. (1/5/2020)
- Section 3.A, page 264. In the first sentence of the proof of Theorem 3A.3 change Ker i_{n-1} to Ker(i_{n-1} tensor the identity). (12/29/2019)
- Section 3.C, page 285, lines 7-8. As originally written, the definition of the coproduct in the tensor product of two Hopf algebras was not given with sufficient care since it neglected the interchange of the middle two factors of a fourfold tensor product, which involves the usual sign change in graded commutativity, For a corrected statement see the pdf errata file or the online version of the book. (9/15/2018)
- Section 3.E, page 304, line 11 of Example 3E.2. Two typos: add absolute value signs to change z_i^2 to |z_i|^2, and change the angle pi to pi/2. (5/29/20)
- 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. (1/7/2018)
- Section 4.1, page 352. Added a simple argument that CW approximations are unique up to homotopy equivalence and deleted this statement from the later Corollary 4.19. Also made small revisions in the bottom half of this page and the top of the next page. (1/15/2020)
- Section 4.1, page 358. In Exercise 5 change the coset alpha H to H alpha. [Explanation: I usually think of the relative pi_1 as homotopy classes of paths from the basepoint to the subspace, but the actual definition on page 343 uses paths in the opposite direction, from points in the subspace to the basepoint.] (6/2/21)
- Section 4.1, page 358, Exercise 9, first line. Change i^* to i_*. (6/2/21)
- Section 4.2, pages 372-373. The proof of the general Hurewicz theorem has been rewritten. (1/19/2020)
- 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)
- Section 4.3, page 409, last paragraph. Made the definition of a fibration sequence more explicit and added the alternative name ``Puppe sequence''. (1/10/2020)
- Section 4.3, page 410, third line above Proposition 4.67. Typo: Change X_n to G_n. (3/17/2020)
- Appendix, page 529, line before Corollary A.12. The reference should be to Corollary A.9 instead of Corollary A.10. (2/10/2021)
- Appendix, page 529. In the paragraph after Corollary A.12 the information about simplicial structures on manifolds has been updated. (1/22/2020)
- Appendix, pages 529-533. A few minor revisions have been made in the section on the compact-open topology for the sake of clarity, expanding this section by half a page. (3/8/2019)
- Index entry for projective space: quaternion. The first reference should be to page 222, not 214. (1/6/2020)