Corrections to the seventh printing of Algebraic Topology

Last updated October 31, 2009.

Below is a list of corrections, clarifications, and comments for the seventh printing of the book (2006). 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.

• Chapter 0, page 14. The discussion of the homotopy extension property in the middle of this page skims over a somewhat delicate question in point-set topology, whether a function defined on the union of X x {0} and A x I is continuous if its restrictions to X x {0} and A x I are each continuous. This is true if A is closed in X, which covers most applications. The online version of the book gives a corrected version of the argument. The trickier case that A is not assumed to be closed has been added to the Appendix. (4/5/09)
• Chapter 0, page 19, Exercise 21. The space X should be assumed to be Hausdorff. For a more general version, let X be a connected quotient space of a finite set of disjoint 2-spheres obtained by identifying finitely many finite sets of points. (9/25/08)
• Section 1.1, page 32, third paragraph. The reference should be to Corollary 2.15 instead of 2.11. (8/21/07)
• Section 1.1, page 32, last paragraph. The reference should be to Corollary 2B.7 instead of Proposition 2B.6. (9/25/08)
• Section 1.1, page 36, line 6. The reference should be to Theorem 2.26 instead of 2.19. (11/12/08)
• Section 1.2, page 55, line 1. A comment: The reduced suspension depends on the choice of basepoint, so the statement is that C is the reduced suspension of CX with respect to a suitable choice of basepoint. (7/23/06)
• Section 1.3, page 57, third-to-last line. Change Koenig to König, to agree with the spelling in the Bibliography and in the original source itself. (8/21/07)
• Section 1.3, page 61, At the end of the next-to-last line of the proof of Proposition 1.32 there should be a dot (denoting composition of paths) between the symbols g_1 and g_2 bar. (9/25/08)
• Section 1.3, page 63. Typo in the next-to-last line of the third-to-last paragraph: "simply-connected" has two n's, not three. (2/21/09)
• Section 1.3, page 69, second and third lines of last paragraph. It should say "assuming that X is path-connected, locally path-connected, and semilocally simply-connected". (10/27/06)
• Section 1.B, page 94. In the middle of the page, change the sentence that begins "To see this" so that it reads "To see this, note that p : \tildë A \to A is a covering space, so we have injective maps pi_1(\tildë A) \to pi_1(A) \to pi_1(X) whose composition factors through pi_1(\tildë X) = 0, hence pi_1(\tildë A) = 0." (4/11/09)
• Section 1.B, page 94, seventh line up from the bottom. Change A to p^{-1}(A). (4/16/09)
• Section 1.B, page 96, Exercise 9. Add the hypothesis that all the edge homomorphisms are injective. (4/16/09)
• Section 2.1, page 120, line -12. Change w_i and w_j to w_j and w_k. (8/28/08)
• Section 2.1, page 123, line -3. Add a period at the end of this line. (8/28/08)
• Section 2.1, page 127. Typo in the line just above the large commutative diagram: sequences. (4/16/09)
• Section 2.2, page 136. See the online version of the book for slightly improved phrasing of the proof of Proposition 2.30. (2/17/09)
• Section 2.2, page 141, two lines above Example 2.36. There should be a tilde over H_{n-1}, though it doesn't really matter since we are in the case n>1. (11/12/08)
• Section 2.2, page 158, Exercise 30, line 2. The numeral 1 that occurs in the label on a map in the exact sequence should be in the blackboard bold font. (11/12/08)
• Section 2.2, page 158, Exercise 34. Replace the sentence in brackets by "The singular complexes S(X) and S(A) can be used to reduce to the case of CW pairs (X,A)." The reason for this change is that the proof given in the book for the isomorphism mentioned in the original sentence in brackets used the long exact sequence of a pair, so there is a danger of circular reasoning here. For CW pairs (X,A) the isomorphism can be proved without using a long exact sequence, and the general case can be reduced to this special case by using the singular complexes S(X) and S(A). (6/21/09)
• Section 2.B, page 176, Exercise 3. A better hint would be to glue two copies of (D^n,D) to the two ends of (S^{n-1}x I,S x I) to produce a k-sphere in S^n and then look at a Mayer-Vietoris sequence for the complement of this k-sphere. (The hint originally given leads to problems with the point-set topology hypotheses of the Mayer-Vietoris sequence.) (6/15/06)
• Chapter 3 Introduction, page 189, line 21. Change the minus sign in psi - delta phi to an equals sign. (8/28/08)
• Section 3.2, page 208. In the last sentence of the first paragraph on this page (this is the sentence referring to Theorem 3.14) it might be a good idea to add, for the sake of clarity, the phrase "assuming that the coefficient ring itself is commutative" at the end of the sentence. (6/21/09)
• Section 3.2, page 210, fifth line of Example 3.11. Insert the word "of" following "generator". (12/9/08)
• Section 3.2, page 210, last line. The comma in the term H^n(I x Y,R) should be a semicolon. (6/6/08)
• Section 3.2, page 213, third paragraph, third line. Change P^n - {0} to P^n - {p}. (4/20/06)
• Section 3.2, page 216, first line. The comma in the term C^l(X,R) at the end of the line should be a semicolon. (6/6/08)
• Section 3.2, page 225, lines 15 and 17. Typo: Change the superscript i + j on x to a subscript. (5/31/09)
• Section 3.2, page 227, first sentence. The reference to the 1980 paper of Adams and Wilkerson is incorrect. In fact the proof of this fundamental result has only been completed recently in a paper of K. Andersen and J. Grodal, The Steenrod problem of realizing polynomial algebras, Journal of Topology 1 (2008), 747-760. (5/20/06)
• Section 3.2, page 228. The algebraic problem referred to at the end of the first paragraph on this page has been solved. The answer is what one would hope: The simplicial complex C_X is uniquely determined by the cohomology ring H^*(X;Z). In fact this is true with Z_2 coefficients. A similar result holds also in the situation mentioned in the following paragraph, so a subcomplex of a product of n copies of CP-infinity is uniquely determined by its cohomology ring, up to permutation of the factors (and deletion of a CP-infinity factor if none of its positive-dimensional cells are used). The reference is Theorem 3.1 in J. Gubeladze, The isomorphism problem for commutative monoid rings, J. Pure Appl. Alg. 129 (1998), 35-65. (12/1/07)
• Section 3.A, page 262, tenth line from the bottom. Missing prime: b - b' = i(a). (4/11/09)
• Section 3.C, page 284, lines 3-4. I forgot to make the correction here for an error originally found in 2005. It should say "so the terms alpha'_i and alpha''_i in the coproduct formula Delta(alpha) = alpha tensor 1 + 1 tensor alpha+ summation_i alpha'_i tensor alpha''_i must be zero." (6/19/09)
• Section 3.D, page 293, line 16. Insert "finite-dimensional" before "CW structure". (7/7/09)
• Section 3.D, page 297. In the last line the superscript I on beta should be a subscript. (7/7/09)
• Section 3.D, page 300, Proposition 3D.4. In the third line the symbol a_{2k+1} should be something different, such as a'_{2k+1}, to avoid ambiguities. The last sentence in the proposition and the last paragraph in the proof should be changed accordingly. (10/20/09)
• Section 3.D, page 300, thirteenth line from the bottom. The reference should be to Theorem 3D.2 rather than Proposition 3D.2. (4/11/09)
• Section 3.D, page 302, Exercise 1. Add the hypothesis that the CW structure is finite-dimensional. (7/7/09)
• Section 4.1, page 349, line 10. Delete the word "to" preceding "try". (12/9/08)
• Section 4.1, page 359, Exercise 22. Add the word "weakly" before "homotopy equivalent". (3/14/07)
• Section 4.2, page 362, line 19. Replace the superscript n-1 by i-1. (9/22/08)
• Section 4.2, page 371, twelfth line from the bottom. Wrong font for the X near the beginning of this line. Should be italic. (4/11/09)
• Section 4.2, page 371, next-to-last line. Delete the superscript 1 on the X at the end of this line. (4/11/09)
• Section 4.2, page 380. At the end of Example 4.50 replace K(Z,3) by K(Z,4). (12/22/06)
• Section 4.2, page 390, Exercise 15. The Poincaré conjecture has been proved. (4/11/09)
• Section 4.2, page 391, line 5. H_n(X) should be H_{n+1}(X). (2/25/08)
• Section 4.3, page 394, third paragraph. The hypothesis that X be connected is unnecessary. Also, a comment could be added at the end of the paragraph that H^0(X;G) = [X,K(G,0)] and H-tilde^0(X;G) = < X,K(G,0)>. (8/28/09)
• Section 4.3, page 399, middle. The label (4) on the displayed exact sequence can safely be omitted. (4/11/09)
• Section 4.3, page 400, line 6. Change h^n(point) to h_n(point). (3/3/08)
• Section 4.3, middle of page 412. In the definition of the k-invariant the coefficient group should be pi_{n+1}(X) instead of pi_{n+1}(K). (6/15/06)
• Section 4.3, page 417, last line. The reference should be to Lemma 4.7 rather than to an exercise in Section 4.1. (3/12/08)
• Section 4.A, page 425, eleventh and tenth lines from the bottom. Change "octagon" to "octahedron" (4/11/09)
• Section 4C, page 429. In the line preceding the diagram the subscript n+1 on Z should be a superscript. (10/31/09)
• Section 4.D, page 445, eleventh line from the bottom. Change "corollary" to "proposition". (4/11/09)
• Section 4.E, page 448. In the diagram near the bottom of the page all the A's should be in the same italic font. (4/11/09)
• Section 4.H, page 463. Delete the extra period at the end of the first paragraph. (5/20/06)
• Section 4.H, page 464, line 14. The superscript on D should be n rather than m. (6/15/06)
• Section 4.I, page 467, line 9. The inclusion map should be S^1 into the wedge sum S^1 v S^1 rather than just into S^1. (4/11/09)
• Section 4.I, page 468, line -19. There is a missing capital Sigma before J_n(X). (7/7/06)
• Section 4.L, page 488, first sentence of the proof of Proposition 4L.1. The identification H^m(X;G) = <X,K(G,m)> is valid only for m > 0. For m = 0 one has H^0(X;G) = [X,K(G,0)]. (8/28/09)
• Section 4.L, page 488, next-to-last line. It would be better to say that there are no nontrivial cohomology operations that decrease dimension. (8/28/09)
• Section 4.L, page 493. Replace the two sentences immediately preceding Example 4L.5 by the following:
  In particular, d is not equal to -1. The Lefschetz number lambda(f) = 1 + d + ... + d^n = (d^{n+1} - 1)/(d - 1 ) is therefore nonzero since the only integer roots of unity are 1 and -1. The Lefschetz fixed point theorem then gives the result. (6/17/09)
• Section 4.L, page 500, sixth line from the bottom. Change 4K.1 to 4L.1. (4/11/09)
• Section 4.L, page 509. There are sign problems in the proof of Lemma 4L.14. For a corrected version of the argument see the online version of the book. (4/27/09)
• Appendix, page 528, ninth line from the bottom. Add a subscript 2i-1 to the X symbol at the beginning of this line. (4/11/09)
• Index, page 541. In the entry for the Hurewicz theorem the first two page numbers should be 366 and 371. (3/23/08)