Corrections to Algebraic Topology --- last updated March 12, 2008.

Below is a cumulative list of corrections, clarifications, and comments for the first printing of the book. The list can also be downloaded in pdf format, with a few more details in some cases. Many of these corrections have been incorporated into later printings. Undoubtedly more errors remain to be discovered. If you find any, please send them to me for inclusion in this list and to correct future printings.

• Chapter 0, page 9. In the next-to-last paragraph delete the sentence "This viewpoint makes it easy to see that the join operation is associative." Also, in the sentence preceding this one, change the word "regarded" to "constructed". Set-theoretically it is true that join is associative, but there are examples where the topologies on (X*Y)*Z and X*(Y*Z) can be different. This is another instance of how mixing product and quotient constructions can lead to bad point-set topological behavior. For CW complexes the issue can be avoided by using CW topologies, as in the first paragraph at the top of the next page. (12/13/05)
• Chapter 0, page 9, line -11. Replace 0t_i by 0x_i. (12/13/05)
• Chapter 0, page 15, Example 0.15. If you have an early version of this chapter with no figure for this example, then in the next-to-last line of this paragraph change "the closure of X - N" to "X - h(M_f - Z)". [This paragraph was rewritten for later versions, making this correction irrelevant.] (11/30/01)
• Chapter 0, page 17, Proof of 0.19. The fourth line should say (Y,A) has the homotopy extension property, rather than (X,A). (11/30/01)
• In the next paragraph, in the fourth and twelfth lines after the displayed formula for k_t the functions k_tu should be maps from A to X rather than from A to A. (6/20/02)
• Same page, seventh-to-last line. It might be clearer to say "Viewing k_{tu} as a homotopy of k_t | A ..." (4/15/03)
• Also in the proof of 0.19, third-to-last line, the function f_1 should go from X to Y, not X to X. (12/19/02)
• Chapter 0, page 20, Exercise 26, third line: Change (X,A) to (X_1,A). (4/15/03)
• Chapter 0, page 20, Exercise 27. To avoid difficulties with point-set topology, assume not just that f is surjective but that f is in fact a quotient map. (4/15/03)
• Section 1.1, page 30, line 14. Change "paths lifting the constant path at x_0" to "paths lifting constant paths" (1/26/04)
• 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 39, Exercise 16(c). In case it's not clear, the circle A is supposed to be the dark one in the figure, in the interior of the solid torus. (2/4/04)
• Section 1.2, page 46, sixth line from bottom. Repeated "the" -- delete one. (11/28/04)
• Section 1.2, page 53, Exercise 5. Part (b) is simply wrong, and should be omitted. (10/23/02)
• Section 1.2, page 54, Exercise 15. It should be specified that if the triangle T has vertices P, Q, R, then the three edges are oriented as PQ, PR, QR. (9/12/03)
• 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 56, second paragraph. A comment about the definition of a covering space: The way that p^{-1}(U_{\alpha}) could be empty is that it could be the union of an empty collection of open sets homeomorphic to U_{\alpha}. (1/9/06)
• 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 65, line 12. Change "cover space" to "covering space" (4/16/02)
• 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 1.3, page 79, Exercise 3. Add the hypothesis that the covering space map is surjective. (11/1/01)
• Section 1.3, page 79, Exercise 8. The reference should be to Exercise 11 in Chapter 0, not Exercise 10. (10/25/05)
• Section 1.3, page 82, Exercise 33. Change the letter "el" in the formula in the fourth line to d. (8/25/03)
• Section 2.1, page 121, line -9. There is a missing S or two in these equations. See the pdf file of corrections for details. (2/20/02)
• Section 2.1, page 125, Example 2.23. Each occurrence of H_n(S^n) in this example should have a tilde over the H. (11/17/03)
• Section 2.1, page 127. In the discussion of naturality two-thirds of the way down the page the maps alpha, beta, and gamma should be assumed to be chain maps. (12/15/01)
• Section 2.1, page 129, next-to-last paragraph. In each of the first, second, and fourth lines of this paragraph there is a k'(c) that should be k'(c'). (4/27/02, 12/24/04)
• Section 2.2, page 134. The notion of degree is not very interesting when n = 0, so it may be best to exclude this case from the definition to avoid thinking about trivialities and whether there should be a tilde over H_n. (11/17/03)
• Section 2.2, page 135, last line. Add the condition n > 0 to avoid having to speak of reduced homology later in this paragraph. (3/10/02)
• Section 2.2, page 137, line 6. Change "stretching" to "shrinking." (3/10/02)
• Section 2.2, page 152. In the exact sequence at the top of the page delete the final 0 and the arrow leading to it. (5/7/02)
• Section 2.2, page 156, Exercise 13. The second half of part (b) should say that the only subcomplex A for which the quotient map from X to X/A is a homotopy equivalence is the trivial subcomplex consisting of the 0-cell alone. (5/4/02)
• Section 2.3, page 164. There is evidently something wrong with the syntax of the long sentence beginning on line 8 of this page, the second example of a functor. The simplest correction would be to change the word "assigns" to "assigning" in line 8. Perhaps a better fix would be to break this long sentence into two sentences by putting a period at the end of line 9 and then starting a new sentence on line 10 with "This is a functor from the category ...". (1/5/02)
• Section 2.B, page 173. In the second paragraph after Theorem 2B.5 the historical comments are in need of corrections. Frobenius' theorem needs the hypothesis that the division algebra has an identity element, and Hurwitz only proved that the condition |ab| = |a||b| implies the dimension must be 1, 2, 4, or 8. Other people later refined this to show that, in the presence of an identity element, the only examples satisfying |ab| = |a||b| are the classical ones R, C, H and O. The proofs of these results are algebraic. Topological methods only come into play when the condition |ab| = |a||b| is dropped. (2/23/04)
• 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)
• Section 2.C, page 180. In the line preceding the proof of 2C.3 the S^3 should be S^4. Also, in the line above this the reference should be to Example 4L.4 rather than to an exercise in section 4K. (11/13/03)
• Section 2.C, page 180. The last sentence on this page continuing onto the next page is somewhat unnecessary since the fact that K is a subdivision of L implies that its simplices have diameter less than epsilon/2. (12/19/02)
• Chapter 3 Introduction, page 187, fourth-to-last line. Change "homology group" to "cohomology group". (10/24/03)
• Section 3.1, page 198, line 20. There are two missing phi's. It should read phi(boundary sigma) = phi(sigma(v_1)) - phi(sigma(v_0)) = 0. (12/1/03)
• Section 3.1, page 202 line 5. Change H^n(X,A) to H^n(X,A;G). (10/11/04)
• Section 3.2, page 213, third paragraph, third line. Change P^n - {0} to P^n - {p}. (4/20/06)
• Section 3.2, page 215. In the statement of Theorem 3.14 change "with" to "when". (12/19/02)
• Section 3.2, page 217, sixth to last line. Change "a special case of the former if 2 is not equal to 0 in R" to "a consequence of the former if R has no elements of order 2". (11/16/04)
• Section 3.2, page 218, last line of second paragraph: Change the first Y to X, so that the formula becomes the tensor product of the cohomology of X with the cohomology of Y. (2/20/02)
• 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 3.2, page 228. Change Macauley to Macaulay (3 times). Also in the Index, page 540, it should be Cohen-Macaulay. (4/15/03)
• Section 3.2, page 229, Exercise 4. The reference should be to Exercise 3 in Section 2.C. (11/13/03)
• Section 3.2, page 229, Exercise 5. Change this to: Show the ring H^*(RP^{infinity};Z_{2k}) is isomorphic to Z_{2k}[alpha,beta]/(2alpha,2beta,alpha^2 - k beta) where |alpha|=1 and |beta|=2. [Use the coefficient map Z_{2k} ---> Z_2 and the proof of Theorem 3.12.] (5/28/04)
• Section 3.2, page 230. In the next to last line of Exercise 14 the exponent on alpha should be 2n+1 instead of n+1. (5/28/04)
• Section 3.2, page 230, Exercise 17. This can in fact be done by the same method as in Proposition 3.22, although the details are slightly more complicated. For a write-up of this, here is a pdf file. (11/28/03)
• Section 3.3, page 234, line 7. Change "neighborhood of A" to "neighborhood of the closure of A" (11/13/03)
• Section 3.3, page 236. In the sixth line of the long paragraph between Theorem 3.26 and Lemma 3.27 change the phrase "for B any open ball in M" to "for B any open ball in M containing x." (11/21/01)
• Section 3.3, page 239, next-to-last line, change "(k - l)-simplex" to "(k - l)-chain. (11/30/01)
• Section 3.3, page 241. In the ninth-to-last line change "cycle" to "cocycle". (11/13/03)
• Section 3.3, page 242. In line 5 of the subsection Cohomology with Compact Supports change "chain group" to "cochain group." (3/19/02)
• Section 3.3, page 245. At the end of the first paragraph on this page it is stated that inclusion maps of open sets are proper maps, but this is not generally true, so this statement should be deleted. Proper maps do induce maps on cohomology with compact supports, but the proof of Poincar\'e duality uses induced maps of a different sort going in the opposite direction from what is usual for cohomology, maps H_c^i(U;G) \to H_c^i(V;G) associated to inclusions U \subsetof V of open sets in the fixed manifold M. (11/13/03)
• Section 3.3, page 245. In the diagram in the middle of this page the two vertical arrows are pointing in the wrong direction in the first printing of the book. This was corrected in the second printing. (11/26/01)
• Section 3.3, page 248. In the next-to-last line of item (1) in the proof of Poincare Duality, change "the cocycle taking" to "a cocycle phi taking" (11/26/01)
• Section 3.3, page 249, line 12. Change H^{i-1} to H^{i+1} (9/12/03)
• Section 3.3, page 249. In the line above the commutative diagram two-thirds of the way down the page there are a couple missing symbols in the two Hom groups. What is missing is ,R) at the end of each Hom term. (11/13/03)
• Section 3.3, page 251, last line. There is a missing parenthesis following the second H^j (9/12/03)
• Section 3.3, page 253. In the last paragraph of the proof of Proposition 3.42 it might be better to replace all the subscripts i by k. (12/23/03)
• Section 3.3, page 255, line 5. Omit the coefficient group Z. (It should have been a blackboard bold Z in any case.) (9/12/03)
• Section 3.3, page 256, lines 1-2. Change the superscript 0 to a subscript and change the two superscripts n to n-1. (5/2/05)
• Section 3.3, page 256, line 8. Change "Example 1.26" to "Example 1.24". (11/13/03)
• Section 3.3, page 258, Exercise 8, second line. Delete the second "of". (11/13/03)
• Section 3.B, page 268, tenth-to-last line. Change "homomorphism" to "bilinear map". (11/13/03)
• Section 3.B, page 272, first line. Change "for all i" to "for all n" (4/16/02)
• Section 3.B, page 273. In the displayed equations near the bottom of the page the last (-1)^i should be deleted. (9/12/03)
• Section 3.B, page 276, Corollary 3B.2 (which incidentally should have been numbered 3B.8). The isomorphism in this corollary is obtained by quoting the Künneth formula and the universal coefficient theorem, whose splittings are not natural, so the isomorphism in the corollary need not be natural as claimed. However there does exist a natural isomorphism, obtainable by applying Theorem 4.59 later in the book. (11/13/03)
• Section 3.B, page 280, next-to-last line before the exercises. Change Delta T to T Delta. (9/12/03)
• Section 3.B, page 280, Exercise 5, lines 2 and 3. The slant products should map to the homology and cohomology of X rather than Y. (9/1/2004)
• Section 3.C, page 281. In the last two lines of the next-to-last paragraph, change it to read "... compact Lie groups O(n), U(n), and Sp(n). This is explained in §3.D for GL_n(R), and the other two cases are similar." (11/13/03)
• Section 3.C, page 282, tenth line up from the bottom. Change SP_{n+1} to SP_{n+1}(X). (10/11/04)
• Section 3.C, page 283. The summation displayed in line 14 is not sufficiently general. It should be a sum of terms alpha'_i tensor alpha''_i where |alpha'_i| > 0 and |alpha''_i| > 0. There are four other places in this section where a similar correction needs to be made: in item (2) later on this page, in lines 3-4 on the next page, and in items (2) and (4) on page 290. See the pdf errata file for more details. (4/5/05)
• Section 3.C, page 286, Example 3C.5, third line. Change 2i to ni. (9/12/03)
• Section 3.C, page 286, eleventh line up from the bottom. Add the condition i > 0 for the statement about alpha_{p^i} being nonprimitive in the divided polynomial algebra. (11/13/03)
• Section 3.C, page 291, Exercise 3. Assume the H-space multiplication is homotopy-associative (i.e., associative up to homotopy). (11/21/03)
• Section 3.C, page 291, Exercise 9. Add the hypothesis that X is connected. (11/13/03)
• Section 3.C, page 291, Exercise 10, part (c). Assume that a_n and b_n are nonzero. (11/13/03)
• Section 3.D, page 295. In the text to the left of the figure change P^n to P^{n-1}. (9/12/03)
• Section 3.F, page 314, lines 9-10. The finite expressions b_n ... b_1b_0 correspond just to nonnegative integers. (10/29/02)
• Section 3.F, page 315, next-to-last line of first paragraph. Change H^n to h^n. (9/12/03)
• Section 3.G, page 322, line 5. The first X in this line should have a tilde. (10/11/04)
• Section 3.G, pages 326-327. The list of Lie groups whose classifying spaces have polynomial mod p cohomology rings is incomplete for the prime 2. Perhaps the best way to describe the situation would be to restrict the discussion to odd primes up until the last paragraph in this section, and then enlarge the final table for the prime 2 to include the missing examples. Among these are G_2 with polynomial generators in degrees (4,6,7), Spin(7) with generators in degrees (4,6,7,8), Spin(8) with generators in degrees (4,6,7,8,8), Spin(9) with generators in degrees (4,6,7,8,16), F_4 with generators in degrees (4,6,7,16,24), and PSp(2n+1) with generators in degrees (2,3,8,12,...,8n+4). Here PSp(n) is the quotient of Sp(n) by its center, which consists of just plus or minus the identity matrix. I have been told there may be other examples as well, and I will post these here when I get a more complete list from the experts on this subject. (Note that for p=2 the term "degree" means the actual cohomological dimension, whereas for odd primes it meant half the cohomological dimension.) (8/25/05)
• Section 3.H, page 332, line -9. Change "Bockstein" to "change-of-coefficient". (10/6/03)
• Section 3.H, page 334, line 2. Missing parenthesis in C^n(X;E). (7/18/02)
• Section 3.H, page 334, line following Proposition 3H.5. Repeated "the" -- delete one. (11/28/04)
• Section 3.H, page 335. In the statement of Theorem 3H.6, Poincare duality with local coefficients, change the second (or alternatively, the third) occurrence of M_R to R, ordinary coefficients in R rather than local coefficients. For more details see this pdf file. (7/18/02)
• Section 4.1, page 345, line 2. Change (X,B,x_0) to (X,A,x_0). (12/19/02)
• Section 4.1, page 359, Exercise 22. Add the word "weakly" before "homotopy equivalent". (3/14/07)
• Section 4.2, page 361, line 18. Repeated "the" -- delete one. (11/28/04)
• Section 4.2, page 370. The large diagram on this page will only commute up to sign unless the generators alpha are chosen carefully. Commuting up to sign is good enough for most purposes, so this isn't really a big issue. It might be a good exercise to see how to choose generators to make the diagram commute exactly. (3/2/04)
• Section 4.2, page 374. Delete the stray direct sum symbol in the displayed exact sequence. (2/20/02)
• Section 4.2, page 376. In the proof of injectivity of p_* there is an implicit permutation of the last two coordinates of I^n x I when the relative homotopy lifting property is applied. (3/2/04)
• 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 398, line 3. Change SX to SA. (3/10/05)
• Section 4.3, page 399, third paragraph. Change L to K', twice. (9/12/03)
• Section 4.3, page 400, line 6. Change h^n(point) to h_n(point). (This involves changes made after the first printing, so the first printing is unaffected.) (3/3/08)
• Section 4.3, page 409, next-to-last line of next-to-last paragraph. Switch gamma and eta, so that it reads "composing the inverse path of p eta with gamma." (9/12/03)
• 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 4.3, page 419, Exercise 6. It should have been explained how the cross product is defined since we are using coefficients in G rather than a ring. However, instead of using cross products it would be better just to use Exercise 4 to construct the H-space structure and prove the stated properties. The problem could also be expanded to include showing that the H-space structure has a homotopy-inverse. (1/10/06)
• Section 4.3, page 419, Exercise 8. Typo in the second line: The "p" in ps=1 should be a Greek "pi". (1/10/06)
• Section 4.3, page 420, Exercise 13. Small typo: It should begin "Given a map". (12/7/03)
• Section 4.B, page 428, line 6. Typo: Adam' should be Adams' (12/15/01)
• Section 4.C, page 430, third line of Example 4C.2. Insert the word "and" before H_{n+1}(X). (10/13/03)
• Section 4.E, page 450. In the third-to-last line T_n should be T_{u_n}, and again in the next line as well. (3/11/05)
• Section 4.F, page 454, last paragraph. The definition of a cohomology theory associated to a spectrum as a limit doesn't always work for infinite CW complexes, as the wedge sum axiom can fail. The easy way to avoid this problem is to associate an Omega spectrum to a spectrum as described in the first paragraph on tthe next page, and then take the cohomology theory defined by this Omega spectrum. (6/20/02)
• 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)
• Section 4.I, page 470, Exercise 2. In the formula in the first line there are three missing Sigmas before the three K's. Also, in the last line the reference should be to Proposition 4I.3 instead of 4E.3. (1/5/04)
• Section 4.L, page 491, seventh line from the bottom. The exponent n + 4i should be n + 2i. The same correction should be made again on the second line of the next page. (9/12/03)
• Section 4.L. Starting on page 496 and continuing for the rest of this section the name Adem is mistakenly written with an accent. (12/15/01)