Corrections to the seventh printing of Algebraic Topology

Last updated September 25, 2008.

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 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 1.2, 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 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 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.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 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 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 work of K. Andersen and J. Grodal that has yet to be published. (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 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 380. At the end of Example 4.50 replace K(Z,3) by K(Z,4). (12/22/06)
• Section 4.2, page 391, line 5. H_n(X) should be H_{n+1}(X). (2/25/08)
• 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.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 468, line -19. There is a missing capital Sigma before J_n(X). (7/7/06)
• Index, page 541. In the entry for the Hurewicz theorem the first two page numbers should be 366 and 371. (3/23/08)