Corrections to *Algebraic Topology*
--- last updated Septeember 15, 2018.

Below is a cumulative list of corrections, clarifications, and comments for the various printings of the book. The list can also be downloaded in pdf format, with a few more details in some cases. Most 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 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. (3/11/2016)
- Chapter 0, page 9, line 12. Change "lines" to "line". (9/18/12)
- 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 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 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 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)
- 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 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)
- 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 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 1.1, 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 53, Exercise 6. Add the assumption that the discrete subspace of R^n is closed. The result still holds without this condition but the proof is considerably more complicated. (3/17/15)
- 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 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 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 "ell" in the formula in the fourth line to d. (8/25/03)
- 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 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 $ \x $ 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 120, line -12. Change w_i and w_j to w_j and w_k. (8/28/08)
- 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 123, line -3. Add a period at the end of this line. (8/28/08)
- 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 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. If you have an early printing of the book where the next-to-last commutative diagram on this page is a small diagram consisting of two short exact sequences joined by vertical maps alpha, beta, and gamma, then add the hypothesis that these maps are chain maps, commuting with boundary homomorphisms. If you have a later printing with a large three-dimensional commutative diagram which includes the boundary maps as well as the maps alpha, beta, and gamma, then nothing more needs to be added. However, in the line preceding this large diagram there may be a typo in the word "sequences" in your printing of the book.
- 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 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 137, line 6. Change "stretching" to "shrinking." (3/10/02)
- 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 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 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 2.2, 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 2.2, 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 2.3, 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 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 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, line -11. Typo: The formula involving tau should be just tau(fr) = tau(f), without the star subscripts. (3/4/2017)
- 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)
- 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 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 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 202 line 5. Change H^n(X,A) to H^n(X,A;G). (10/11/04)
- 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 215. In the statement of Theorem 3.14 change "with" to "when". (12/19/02)
- 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 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 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 3.2, 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. (This paragraph has been revised in later printings of the book, so this correction is no longer relevant.) (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 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 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 250, statement of Corollary 3.39. The condition on alpha should be that it generates an infinite cyclic summand, not that it has infinite order and is not a proper multiple of another element. The condition of generating an infinite cyclic summand is what is used in the proof, and it has the advantage of converting the corollary into an "if and only if" statement. There are examples showing the corrected condition is strictly stronger than the condition originally stated. (12/2/14)
- Section 3.3, page 251, last line. There is a missing parenthesis following the second H^j (9/12/03)
- 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 3.3, 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 257, lines 12-14. The assertion about Cech cohomology satisfying a stronger form of excision holds for compact pairs but not in general. Perhaps the easiest correction here is simply to delete the last half of this sentence beginning with "and indeed". (7/17/13)
- Section 3.3, page 258, Exercise 8, second line. Delete the second "of". (11/13/03)
- Section 3.A, page 262, tenth line from the bottom. Missing prime: b - b' = i(a). (4/11/09)
- 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 second (-1)^i in the last line should be deleted. Thus the coefficient of the term preceding the final 0 should be 1. (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 (3) on page 290. See the pdf errata file for more details. (4/5/05)
- Section 3.C, page 284. I forgot to make the correction on page 284 mentioned just above, so now I have finally made this correction in the online version of the book. (6/19/09)
- 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. (9/15/18)
- 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 293, line 16. Insert "finite-dimensional" before "CW structure". (7/7/09)
- 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.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.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.F, page 319. The proof of Proposition 3F.12 originally given was incomplete. (The gap occurred in the third paragraph on page 319 where the possibility of torsion of order relatively prime to p was overlooked.) The online version of the book contains a corrected proof. (5/31/13)
- 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 333, line 13. Change "bundles of groups" to "bundles of abelian groups". (4/14/12)
- Section 3.H, page 334, line 2. Missing parenthesis in C^n(X;E). (7/18/02)
- 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 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 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 345, line 2. Change (X,B,x_0) to (X,A,x_0). (12/19/02)
- 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. (1/7/18)
- 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, pages 350-351. The statement and proof of Lemma 4.10 have been revised a couple times. Most recently, the statement was revised again in October 2012 to say explicitly that the homotopy takes f^{-1}(e^k) to e^k at all times. The proof gives this additional property, and this property is needed when Lemma 4.10 is used in the proof of Theorem 4.23, Case 1, later in the chapter. (10/20/12)
- 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)
- Section 4.2, page 361, line 18. Repeated "the" -- delete one. (11/28/04)
- 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 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 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 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 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 4.2, page 389, Exercise 11. There are counterexamples to the second half of this problem as originally stated, which involved an analog of the first half with pi_2 replaced by its quotient pi'_2. The current online version of this exercise includes such a counterexample. (6/28/15)
- 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 25. The CW complex X is assumed to be connected, as is implicit in the notation pi_n(X) without a basepoint. (1/6/10)
- 4.2, page 391, Exercise 27. This exercise can be done directly from the definition of relative homotopy groups, so it should really be in section 4.1. (4/29/15)
- 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 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 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). (This involves changes made after the first printing, so the first printing is unaffected.) (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, 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). (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 4.3, 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.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 4.B, page 428, line 6. Typo: Adam' should be Adams' (12/15/01)
- Section 4.C, page 429. In the line preceding the diagram the subscript n+1 on Z should be a superscript. (10/31/09)
- Section 4.C, page 430, third line of Example 4C.2. Insert the word "and" before H_{n+1}(X). (10/13/03)
- 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.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, 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.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.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 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.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 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, 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. Starting on page 496 and continuing for the rest of this section the name Adem is mistakenly written with an accent. (12/15/01)
- Section 4.L, page 500, sixth line from the bottom. Change 4K.1 to 4L.1. (4/11/09)
- Section 4.L, page 501, line 15. In the displayed formula the signs on the two occurrences of the index j in the exponents should be reversed. Thus the first exponent should be 2n-k+j-1 and the second exponent should be n-j. The same reversal of signs on j should be made in the analogous formula involving Steenrod powers near the end of this paragraph. (9/23/12)
- 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 530, Proposition A.14. The definition of local compactness we are using here is that each neighborhood of each point contains a compact neighborhood of the point. This follows the general pattern described on page 62, but it is stronger than the more common definition which is that each point has at least one compact neighborhood. For Hausdorff spaces the two definitions agree. (12/5/12)
- 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)