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)