Corrections To Introduction To Topological Manifolds
Corrections to
Introduction to Topological Manifolds
by John M. Lee
January 5, 2010
Changes or additions made in the past twelve months are dated.
• Page 29, proof of Lemma 2.11: Replace the first sentence of the proof by the following: “It suffices
to show that B satisfies the two defining conditions for a basis, for then the fact that B consists of
open sets guarantees that the topology generated by B is contained in the given topology on X, and
conversely the hypothesis together with Lemma 2.10 implies that every open subset of X belongs to
the topology generated by B.”
• Page 29, paragraph before Exercise 2.15: Instead of “the topologies of Exercise 2.1,” it should
say “some of the topologies of Exercise 2.1.”
• Page 30, last sentence of the proof of Lemma 2.12: Replace U by f −1(U ) (three times).
• Page 30, first paragraph in the “Manifolds” section: Delete the sentence “Let X be a topological
space.”
• Page 38, Problem 2-16(b): Replace part (b) by “Show that for any space Y , a map f : X → Y is
continuous if and only if pn → p in X implies f (pn) → f (p) in Y .”
• Page 38, Problem 2-18: This problem should be moved to Chapter 3, because Int M and ∂M are
to be interpreted as having the subspace topologies. Also, for this problem, you may use without proof
the fact that Int M and ∂M are disjoint.
• Page 40, last line of Example 3.1: Replace “subspace topology on B” by “subspace topology on
C.”
• Page 45, line 15: Change Sn to n
S .
• Page 47, line 5 from bottom: Replace “next lemma” by “next theorem.”
• Page 51, proof of Proposition 3.13, third line: f1(U1), . . . , fk(Uk) should be replaced by
f −1(U
(U
1
1), . . . , f −1
k
k ).
• Page 51, proof of Proposition 3.14, last sentence: Replace “the preceding lemma” by “the
preceding proposition.”
• Page 52, first paragraph after Exercise 3.8: In the first sentence, replace the words “surjective
and continuous” by “surjective.” Also, add the following sentence at the end of the paragraph: “It is
immediate from the definition that every quotient map is continuous.”
• Page 52, last paragraph: Change the word “quotient” to “surjective” in the first sentence of the
paragraph.
• Page 53, line 1: Change the word “quotient” to “surjective” at the top of the page.
• Page 53, Lemma 3.17: Add the following sentence at the end of the statement of the lemma: (More
precisely, if U ⊂ X is a saturated open or closed set, then f |U : U → f (U ) is a quotient map.)
• Page 57, second line after the first displayed diagram: Replace the phrase “Y with the given
topology is homeomorphic to Y with the quotient topology” with “the identity map is a homeomor-
phism between Y with the given topology and Y with the quotient topology.”
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(1/30/09) Page 60, Example 3.35(a), line 4: Change “by a linear transformation” to “by an invertible linear
transformation.”
• Page 62, Problem 3-1: The second part of the problem statement is false. Change the problem to
the following: “Show that a finite product of open maps is open; give a counterexample to show that
a finite product of closed maps need not be closed.”
• Page 62, Problem 3-4: Add: “[Hint: For the unit ball in
n
n
n
R , consider the maps πi ◦ σ−1 : R
→ R
for 1 ≤ i ≤ n, where σ is stereographic projection and π
n+1
n
i is the projection from R
to R that omits
the ith coordinate.]”
• Page 62, Problem 3-6: Insert “nonempty” before “topological spaces.”
• Page 81, first displayed equation: The definition of F should be
x
|x| f −1
,
x = 0;
F (x) =
|x|
0,
x = 0.
• Page 81, first line after the displayed equation: Replace the first sentence by the following:
“Then F is continuous away from the origin because f −1 is, and at the origin because boundedness of
f −1 implies F (x) → 0 as x → 0.”
• Page 81, line 4: Change n
n−1
S
to S
.
• Page 82, line 3 from bottom: Delete “= U ∩ Z” from the sentence beginning “Since U ∩ Z . . . .”
• Page 83, Example 4.30(a): In the first sentence, change “closed” to “open” and change Bε(x) to
Bε(x).
• Page 85, statement of Corollary 4.34: “countable collection” should read “countable union.”
• Page 89, Problem 4-11: Insert “taking ∞ to ∞” after “f ∗ : X∗ → Y ∗.”
• Page 94, Example 5.3, second line: Change “Figure 5.3” to “Figure 5.4.”
(1/5/10) Page 96, line 14 from the bottom: Change f to f0 (twice).
• Page 96, Exercise 5.5: Insert the words “isomorphic to” before “the vertex scheme.”
• Page 99, Lemma 5.4: Replace part (d) by
(d ) For any topological space Y , a map F : |K| → Y is continuous if and only if its restriction to |σ|
is continuous for each σ ∈ K.
(1/5/10) Page 100, first line after the end of the proof: Insert “X” after “topological space.”
• Page 102, line 13 from bottom: Replace the two sentences beginning with “To prove this . . . ” by
the following: “To prove this, suppose the contrary. Then there is some edge e ⊂ Gn such that either
Int e ∩ e or e ∩ Int e is nonempty. Since Int e and Int e are open subsets of M , it follows in either case
that Int e ∩ Int e = ∅.”
• Page 103, Proposition 5.11: In the statement of the proposition, change “simplicial complex” to
“1-dimensional simplicial complex.”
• Page 105, line 2: Replace 1977 by 1952 and [Moi77] by [Moi52].
• Page 106, line 3 from bottom: Replace “even” by “odd.”
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• Page 111, Figure 5.12: In S(SK), the points inside the small triangles should be at the intersections
of the three medians.
• Page 114, Problem 5-2: Replace the statement of the problem by: “Let K be an abstract simplicial
complex. For each vertex v of K, let St v (the open star of v) be the union of the open simplices Int |σ|
as σ ranges over all simplices that have v as a vertex; and define a function tv : |K| → R by letting
tv(x) be the coefficient of v in the formal linear combination representing x.
(a) Show that each function tv is continuous.
(b) Show that St v is a neighborhood of v, and the collection of open stars of all the vertices is an
open cover of |K|.”
• Page 114, Problem 5-3: Delete the phrase “and locally path connected.”
• Page 114, Problem 5-5: Insert the words “isomorphic to” before “the vertex scheme.”
• Page 120, Statement of Proposition 6.2(a): Replace x ∈ ∂ 2
2
B by (x, y) ∈ ∂B .
• Page 125, line 2: Insert the following sentence just before “This shows . . . ”: “It is easy to check that
α is a bijective open map, and therefore a homeomorphism.
• Page 126, Proposition 6.6: Add the hypothesis that n ≥ 2.
• Page 127, second line from bottom: Change B2(0) to B2(0)
{0}.
• Page 131, Part 1 of the definition of the geometric realization: After “sides of length 1,”
insert “equal angles,”.
• Page 135, proof of Proposition 6.11: Change S to M and S to M in the fifth line of the second
paragraph of the proof, and again in the fifth and sixth lines of the third paragraph. [Here M and M
are supposed to denote the geometric realizations of various surface presentations.]
• Page 135, middle of the page: After “and whose restriction to each Pi is a homeomorphism,” add
the words “onto its image.”
• Page 136, line 8 from bottom: Change the surface presentation in that line to S1, S2, a, b, c |
W1c−1b−1a−1, abcW2 .
• Page 138, Theorem 6.14: The first statement of the theorem should begin “Any surface presentation
of a connected surface is equivalent . . . .” The last statement should begin “Therefore, every nonempty
connected compact surface . . . .”
• Page 139, proof of the classification theorem: Replace the first sentence of the proof with “Let
M be the compact surface determined by the given presentation.”
(9/23/09) Page 140, line 11: Change “that goes from a v vertex to a vertex in some other equivalence class”
to “that connects a v vertex with a vertex in some other equivalence class.”
• Page 140, line 14: Change “Step 3” to “Step 2.”
• Page 143, first line after the proof of Prop. 6.18: Change “allows” to “allow.”
• Page 149, Example 7.3: The first line should read “Define maps f, g :
2
R → R by . . . .”
• Page 156, Figure 7.7: The labels I × I, F , and X should all be in math italics.
• Page 156, Exercise 7.2: Change the first sentence to “Let X be a path connected topological space.”
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(9/1/09) Page 157, Proposition 7.13: In the statement of the proposition and in the second line of the proof,
change “a map” to “a continuous map.”
• Page 159, second line from bottom: “induced homeomorphism” should read “induced homomor-
phism.”
• Page 160, Proposition 7.18: In the statement and proof of the proposition, change (ιA)∗ to (ιA)∗
three times (the asterisk should be a subscript).
(8/22/09) Page 166, line 10: Change “space” to “group.”
• Page 174, proof of Lemma 7.35: Change the word “maps” to “morphisms” (twice). Also, in the
second-to-last line of the proof, change “Theorem 3.10” to “Theorem 3.11.” (Actually, the last sentence
is misleading, because the proof is not really exactly like that of Theorem 3.11. It would be clearer to
replace the last sentence of the proof by the following: “If we take W = P and fα = πα in the diagram
above, then the diagram commutes with either f ◦ f or IdP in place of f . By the uniqueness part
of the defining property of the product, it follows that f ◦ f = IdP . A similar argument shows that
f ◦ f = IdP .”)
• Page 176, Problem 7-5: Change “compact surface” to “connected compact surface.”
• Page 177, line 3: The formula should read ιβ : Xβ →
X
α
α.
• Page 188, proof of Theorem 8.7: Replace the third sentence of the proof by “If f : I → n
S
is any
loop based at a point in U ∩ V , by the Lebesgue number lemma there is an integer m such that on each
subinterval [k/m, (k + 1)/m], f takes its values either in U or in V . If f (k/m) = N for some k, then
the two subintervals [(k − 1)/m, k/m] and [k/m, (k + 1)/m] must be both be mapped into V . Thus,
letting 0 = a0 < · · · < al = 1 be the points of the form k/m for which f (ai) = N , we obtain a sequence
of curve segments f |[ai−1,ai] whose images lie either in U or in V , and for which f (ai) = N .” Also, in
the last line of the proof, replace “f is homotopic to a path” by “f is path homotopic to a loop.”
• Page 189, proof of Proposition 8.9: In the last sentence of the proof, change the domain of H to
I × I, and change the definition of H to
H(s, t) = (H1(s, t), . . . , Hn(s, t)).
• Page 191, Problem 8-7: In the third line of the problem, change ϕ(γ) to ϕ∗(γ).
• Page 192, line 4: Change the definition of ϕ to ϕ(x) = (x − f (x))/|x − f (x)|.
• Page 199, second-to-last paragraph: In the second sentence, after “a product of elements of S,”
insert “or their inverses.”
(2/9/09) Page 204, line 8 from the bottom: After “coefficients are zero,” insert the sentence “By convention,
the empty set is considered to be independent.”
(2/9/09) Page 206, Example 9.15, first line: Change “any finite group” to “any finite abelian group.”
• Page 208, Problem 9-4(b): Change the first phrase to “Show that Ker f1 ∗ f2 is equal to the normal
closure of Im j1 ∗j2, . . . .” Add the following hint: “[Hint: Let N denote the normal closure of Im j1 ∗j2,
so it suffices to show that f1 ∗ f2 descends to an isomorphism from (G1 ∗ G2)/N to H1 ∗ H2. Construct
an inverse by showing that each composite map Gj → G1 ∗ G2 → (G1 ∗ G2)/N passes to the quotient
yielding a map Hj → (G1 ∗ G2)/N , and then invoking the characteristic property of the free product.]”
• Page 213, proof of Proposition 10.5: In the second sentence of the proof, change {q} to {∗}.
• Page 218, Figure 10.4: In the upper diagram, one of the arrows labeled ai should be reversed.
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• Page 220, second line below the first displayed equation: Change “clockwise” to “counter-
clockwise.”
• Page 227, line 8: Replace R ∗ S by R ∗ S (three times).
• Page 233, last line: Change the last sentence to “This brings us to the next-to-last major subject
in the book: . . . .”
• Page 238, proof of Proposition 11.10, second line: Change “p maps . . . ” to “f maps . . . .”
• Page 239, Example 11.14, last sentence: Change [b] · [a]−1 · [b]−1 to [b] · [a] · [b]−1.
• Page 247, Example 11.23: Add the hypothesis that n > 1.
• Page 248, Example 11.26: Change C
n
n
π (P ) to Cπ (S ).
• Page 248, statement of Proposition 11.27(b): Insert “(with the discrete topology)” after “The
covering group”.
• Page 249, line 5: Change the formula to “p(ϕ(q)) = p(q) = q” (not p).
• Page 253, Problem 11-9: Change “path connected” to “locally path connected.”
• Page 265, Step 4: In the second line of Step 4, replace “as in Step 3” by “as in Step 2.”
• Page 267, Proof of Prop. 12.9, second paragraph: In the last sentence of the paragraph, replace
K ∩ (g · K) = ∅ by K ∩ (g · K) = ∅.
• Page 268, proof of Theorem 12.11: The first and last paragraphs of this proof can be simplified
considerably by using the result of Problem 3-15.
• Page 272, first paragraph: The last sentence should read “It can be identified with a quotient of
α β
the group of matrices of the form
with positive determinant (identifying two matrices if they
β α
differ by a scalar multiple), and so is a topological group acting continuously on
2
B .”
• Page 277, second paragraph from bottom: After the sentence ending “when g and g differ by
a single edge transformation,” insert the following: “An argument similar to that at the beginning of
the proof shows that g · P and g · P intersect in a vertex precisely when g and g differ by a product
of no more than 4n edge pairing transformations.”
• Page 277, line 6 from the bottom: Change (gσ−1, σ(z0)) to (g0σ−1, σ(z0)).
• Page 281, lines 8 through 6 from the bottom: Replace the sentence beginning “To prove this”
by the following: “To prove this, let K ⊂ M denote the union of P together with its images g · P
under the finitely many g ∈ G such that P ∩ (g · P ) = ∅.”
• Page 283, proof of Corollary 12.18, second to last line: Change “Corollary” to “Proposition.”
• Page 284, line 3: Change “Since Cp(X) acts freely and properly on X” to “Since Cp(X), endowed
with the discrete topology, acts continuously, freely, and properly on X”.
• Page 284, last displayed equation: The last U on the right should be U .
• Page 287, line 10: The sentence “Thus (i) corresponds to the rank 1 case” should read “Thus (ii)
corresponds to the rank 1 case.”
• Page 289, Problem 12-5: Replace the statement of the problem by “Find a group Γ acting freely
and properly on the plane such that
2
R /Γ is homeomorphic to the Klein bottle.”
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• Page 290, Problem 12-9: Replace the second sentence by “For any element e in the fiber over the
identity element of G, show that G has a unique group structure such that e is the identity, G is a
topological group, and the covering map p : G → G is a homomorphism with discrete kernel.”
• Page 301, just above the third displayed equation: In the last sentence of the paragraph, replace
Gi,p : ∆p → ∆p × I by Gi,p : ∆p+1 → ∆p × I.
(3/4/09) Page 301, just below the last displayed equation: Change ∆ to ∆p (twice).
• Page 316, first paragraph: Change the fourth sentence to: “For p > 0, if α : ∆
n
p → R
is an affine
p-simplex, set
sα = α(bp) ∗ s∂α
(where bp is the barycenter of ∆p), and extend linearly to affine chains.”
• Page 319, statement of Lemma 13.21: Hn−1 should be Hn−1.
• Page 320, first paragraph: In the last two lines, Hn−1 should be Hn−1 (twice).
• Page 325, second to last displayed equation: Change Hp(K ) to H∆(K ).
p
• Page 327, line 2: Insert “retraction” after “strong deformation.”
• Page 330, paragraph after Exercise 13.4: Replace [Mun75] by [Mun84].
• Page 332, line 1: The first word on the page should be “subgroups” instead of “spaces.”
• Page 333, line 7: Change “coboundary” to “cocycle.”
• Page 334, Problem 13-8: Replace [Mun75] by [Mun84].
• Page 335, Problem 13-12: Add the hypothesis that U ∪ V = X.
• Page 344, Exercise A.7(a): Since this exercise requires the axiom of choice, it should be moved
after exercise A.9.
• Page 360, just before reference [Moi77]: Insert the following reference:
[Moi52]
Edwin E. Moise. Affine structures in 3-manifolds. V. The triangulation theorem and
Hauptvermutung. Ann. of Math. (2), 56:96–114, 1952.
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