Corrections to the seventh printing of Algebraic Topology

Last updated April 14, 2012.

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" should have 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. The original form of this problem was to derive the long exact sequence of homology groups for a pair (X,A) from the Mayer-Vietoris sequence. However, this is hard to do without resorting to some type of circular reasoning, so it seems best to delete this problem. (4/23/11)
• Section 2.B, page 170. In some versions of the book there is a typo in the last line of the proof of Proposition 2B.1. If your copy has h(D^k) in this line, this should be replaced by h(S^k). (Early printings of the book used a different notation here, and the typo was only introduced when the notation was changed.) (3/22/10)
• 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 3.1, page 200. In the diagram that contains the long dashed arrow going diagonally downward there are four occurrences of the letter G. These should be deleted, along with the semicolons that precede them. (4/18/11)
• Section 3.1, page 203, last line. Change the comma in C^n(A+B,G) to a semi-colon. (7/7/10)
• 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 3.3, page 247, lines 1-2. There is a small and somewhat subtle step missing here. For details see the pdf file of corrections. (4/23/11)
• Section 3.3, page 252. In the fourth paragraph, just below the middle of the page, it is stated that every symmetric nonsingular bilinear form occurs as the cup product pairing in a closed simply-connected manifold with miminum homology. This is true in dimensions 4, 8, and 16 but not in other dimensions, where only the even forms are realizable in this way. Certain other forms that are not even are realizable by manifolds with nonminimal homology (such as complex projective spaces), but it doesn't seem to be known whether all forms are realizable. (11/27/09)
• 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 3.D, page 304, line 11. Change the subscript p in Z_p to m. (7/7/10)
• Section 3.H, page 333, line 13. Change "bundles of groups" to "bundles of abelian groups". (4/14/12)
• Section 3.H, page 334, line 5. Typo: Change the G at the end of this line to pi. (4/14/12)
• Section 3.H, page 336, Exercise 5. The assertion that the H^1 group is an infinite direct sum of copies of Z holds only when pi_1(X) is free on two or more generators. When pi_1(X) is infinite cyclic the H^1 group is just a single Z. (1/6/10)
• Section 3.H, page 336, Exercise 6. In the last part of the question add the assumption that X is finite-dimensional. (1/19/10)
• Section 4.1, page 339, second line of last paragraph. The reference should be to section 4.B instead of 4.C. (1/4/10)
• Section 4.1, page 349, line 10. Delete the word "to" preceding "try". (12/9/08)
• Section 4.1, page 349, 13 lines from the bottom. It should perhaps be mentioned that the deformation of f on e^k to make f(e^k) miss the point p will not make f(e^k) intersect any more cells than it intersected before.
• Section 4.1, page 350. The second paragraph has been revised to avoid a possible ambiguity in what was meant by the boundary of K_2. (This was supposed to be the boundary in the sense of point set topology, the intersection of K_2 with the closure of its complement in I^n.) (12/9/11)
• Section 4.1, page 354, eighth line up from the bottom. Delete one of the duplicated words "be". (7/7/10)
• Section 4.1, page 358, Exercise 9, first line. To avoid an abuse of notation, replace pi_0(A,x_0) by i^*(pi_0(A,x_0)). (1/6/10)
• Section 4.1, page 359, Exercise 22. Add the word "weakly" before "homotopy equivalent". (3/14/07)
• 4.2, page 361. The latter part of the paragraph preceeding the figure has been reworded for clarity. See the online version of the book. (Another slight rewording: January 2010) (1/6/10)
• 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)
• 4.2, page 391, Exercise 9. The CW complex X is assumed to be connected, as is implicit in the notation pi_n(X) without a basepoint. (1/6/10)
• 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, page 403. Sixteen lines from the bottom, change z to gamma twice in this line, for notational consistency with the use of gamma earlier in the paragraph. (1/4/10)
• 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). (For consistency, the parentheses surrounding this X can be deleted.) (6/15/06)
• On the line preceding the previous correction, change pi_{n+1} to pi_{n+1}X. (7/7/10)
• 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 4.3, page 418. In the paragraph containing the diagram it should be stated, for the sake of clarity, that F is the fiber of the fibration X ---> Y. (1/4/10)
• Section 4.A, page 422, second and sixth lines from the bottom. Typo in subscripts: It should be Z[pi_1(X)] instead of Z[pi_n(X)]. (1/2/2010)
• 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 4D, page 438, line 13. The tensor product should be over the ring R, so add a subscript R to the tensor product symbol. (10/21/10)
• Section 4.D, page 445, eleventh line from the bottom. Change "corollary" to "proposition". (4/11/09)
• Section 4.D, page 447, exercise 8. In the second line replace Sigma^n B by Sigma^n(B_+) where B_+ is the union of B with a disjoint basepoint. Also, in the last part of the problem the cohomology isomorphism should have both groups being reduced cohomology. (1/4/10)
• Section 4.E, page 448, fourth line after Theorem 4E.1. Change the R to G as the coefficient group. (1/4/10)
• 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.E, page 449, tenth line from the bottom. Change the word "two" to "a few". (There are now three comments -- see the next correction.) (9/17/10)
• Section 4.E, page 450. There is a new item (3) to be inserted following the first paragraph on this page. The content of this new item is that h(X) is a group whenever X is a suspension, and moreover the maps T_u are homomorphisms in this case. Details are given in the pdf file of Corrections, and take about half a page. The reason for adding this new material is that it is implicitly used in the proof of Lemma 4E.4 on page 451, in the fifth sentence of the proof. Once the new item (3) has been added, the sentence in the paragraph preceding Lemma 4E.3 that begins "Note that having a trivial kernel" should be deleted. (In the online version of the book a few minor changes in wording have been made in the proof of Lemma 4E.3 to take advantage of the new item (3).) (9/17/10)
• Section 4.F, page 454, two lines above Proposition 4F.2. Typo: change K_n to X smash K_n, where "smash" is the inverted V symbol. (5/28/11)
• Section 4.G, page 456, thirteenth line up from the bottom. It should be the inclusion of X_0 into X_1, not the inclusion of X_1 into X_1. (7/7/10)
• Section 4.H, page 463, line 4. Delete the extra period at the end of the paragraph. (5/20/06)
• Section 4.H, page 463, line 5. Typo: let us consider. (1/4/10)
• 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.I, page 470, Exercise 3. The lens space should be assumed to be of high dimension. (1/4/10)
• Section 4.j, page 473. At the end of the paragraph containing the commutative diagram, add "by Example 4A.3." (5/28/11)
• Section 4.K, page 480. In the statement of part (b) of Lemma 4K.3, instead of assuming that B has the weak or direct limit topology, assume that each compact set in B is contained in some B_n. (This is to avoid point-set topology issues.) In the electronic version of the book the proof of this lemma has also been revised slightly, clarifying basepoint issues in parts (a) and (b) and simplifying the proof in (c). (9/18/11)
• Section 4.K, page 482. In Example 4K.5 it is the unreduced suspension rather than the reduced suspension that is being used, so to be consistent with the notation elsewhere in the book, each of the five occurrences of the symbol Sigma in this example should be replaced by S. (1/4/10)
• 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 503, line 5. Typo: The word "definition" should be "definitions". (11/27/09)
• Section 4.L, page 503, eighth and ninth lines from the bottom. Change (p-1)n to (p-1)n^2 twice. (1/4/10)
• Section 4.L, pages 504-505, second and third paragraphs of the proof of Theorem 4.12. There is a mistake here since lambda is in fact not additive. Fortunately there is a simple way to deduce additivity of Sq^i from the other axioms, and this argument is now given in the online version of the book. (Correcting this proof has produced changes in the page breaks for pages 502-513.) (11/27/09)
• 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 521. The statement of condition (i) in Proposition A.2 has been revised for clarity, to avoid an implicit dependence on condition (ii). The paragraph following the proposition has been revised accordingly. (5/23/11)
• 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)
• Appendix, page 532, last line of the proof of Proposition A.16. Typo: Change (X^Y)^Z to X^Y. (9/24/11)
• Appendix, page 532. In the first line of the added section on the Homotopy Extension Property change Chapter 1 to Chapter 0, and in the second line the word "certain" is misspelled. (7/7/10)
• Appendix, page 533, fourth-to-last line of the proof of Proposition A.18 (this proposition was only added to the Appendix in 2009). Add a bar over the symbol A. Also the rest of this sentence should say that r_1 is a continuous map to X and X intersect O is open in X. (5/4/10)
• Index, page 541. In the entry for the Hurewicz theorem the first two page numbers should be 366 and 371. (3/23/08)