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Errata list for An Introduction to Algebraic Topology 4th edition.

はじめに

私は数年前に Joseph J Rotman著 『An Introduction to Algebraic Topology』を読んで、代数的トポロジーの勉強をしました。当時日本語で書かれた代数的トポロジーの書籍を何冊か探したのですが、定義や証明が十分に形式的に記述されているものはありませんでした。この本は、非常に定義、証明共に厳密で自習するには最適だったのですが、とにかく誤植が多い印象を受けました。
私が見つけた誤植を全て tex ファイルにして著者に送付したのですが何の response も得られませんでした。そこで、放置するのももったい無いので私のブログで公開しておきます。
これからこの書籍を使って勉強する人の役に少しでも立てばと思います。(詳細はすっかり忘れているので、質問には答えられないと思います)

Errata list

page 53 line 5

 \pi_{1}(R^{1},1) to  \pi_{1}(S^{1},1)

page 74 line -3

 \displaystyle \sum{m_{i}\partial(b.\sigma_{i})} = \sum{m_{i}(b-x_{i})} = (\sum{m_{i}})b-\gamma
to
 \displaystyle \sum{m_{i}\partial(b.\sigma_{i})} = \sum{m_{i}(x_{i}-b)} = \gamma-(\sum{m_{i}})b

 \sum{m_{i}\partial(b.\sigma_{i})} equals to  \sum{m_{i}( (b.\sigma_{i})(e_{1}) - (b.\sigma_{i})(e_{0}))}, and
applying  e_{1}(or  e_{0}) to  b.\sigma means that applying  t=0(or  t=1) to the equation  (b.\sigma)(t) = tb + (1-t)x.

page 77 line -7

 (\lambda^{X}_{1} - \lambda^{X}_{0} - P^{X}_{n-1}\partial_{n})(\sigma)
to
 ({\lambda^{X}_{1}}_{\#} - {\lambda^{X}_{0}}_{\#} - P^{X}_{n-1}\partial_{n})(\sigma)

page 114 line 6

I think that the sentence "It is easy to see that both definitions of  \text{Sd}_{n}(\sigma) agree when X is convex."
should be deleted or be changed to
"It is easy to see that both definitions of  \text{Sd}_{n}(\sigma) agree when X is convex and  \sigma is affine."
(when  \sigma is called  \textit{affine} will be defined in latter page!)

According to the definition of barycentric subdivision when X is a convex set,
\[\text{Sd}_{n}(\sigma) = \sigma(b_{n}).\text{Sd}_{n-1}(\partial\sigma) =
\begin{cases}
\sigma(b_{n}) & \text{if}\ t_{0} = 1 \\
t_{0}\sigma(b_{n}) + (1 - t_{0})\text{Sd}_{n-1}(\partial\sigma) \left( \frac{t_{1}}{1-t_{0}}, \dots ,\frac{t_{n+1}}{1-t_{0}} \right) & \text{if}\ t_{0} \neq 1
\end{cases} \].

On the other hand, according to the definition of barycentric subdivision when X is any space,
\[\text{Sd}_{n}(\sigma) = \sigma_{\#}\text{Sd}_{n}(\delta^{n}) =
\begin{cases}
\sigma(b_{n}) & \text{if}\ t_{0} = 1 \\
\sigma \left( t_{0}b_{n} + (1 - t_{0})\text{Sd}_{n-1}(\partial\delta^{n}) \left( \frac{t_{1}}{1-t_{0}}, \dots ,\frac{t_{n+1}}{1-t_{0}} \right) \right) & \text{if}\ t_{0} \neq 1
\end{cases}\].
These two do not coincide if  \sigma is not  \textit{affine}.

page 118 line -4

 D = d h_{\#}^{-1} q_{\#}
to
 D = d h_{*}^{-1} q_{*}

page 157 line 14

"the first is 18  \times 27 and the second is 27  \times 9"
to
"the first is 27  \times 18 and the second is 9  \times 27"

page 159 line -12

 z - z' \in B_{q}
to
 z - z' \in B_{q+1}

page 183 line 6

 ki = - ki = j
to
 ki = -ik = j

page 189 line -6

 \{\nu(\beta)\} \times \bar{W} \subset \bar{U}
to
 \{\nu(\beta)\} \times \bar{W} \subset U'

page 191 line -6

left top corner of 2nd diagram  U - \{0\}
to
 U' - \{0\}

page 213 line -13

 i\colon (X^{k-1},\emptyset) \hookrightarrow (X^{k},X^{k-1})
to
 i\colon (X^{k-1},\emptyset) \hookrightarrow (X^{k-1},X^{k-2})

page 266 line -18

"Since  \Delta^{p} \times \Delta^{p} is convex, we see that the model  (\Delta^{p},\Delta^{p}) is F-acyclic."
to
"Since  \Delta^{p} \times \Delta^{q} is convex, we see that the model  (\Delta^{p},\Delta^{q}) is F-acyclic."

page 276 line 14

 vw^{-1} = k^{-1}h
to
 vw^{-1} = h^{-1}k

page 277 last line

"Consider the diagram of continuous maps"
to
"Consider the  \textbf{commutative} diagram of continuous maps"

page 288 line 15

"there exits a  \textbf{unique} continuous  h\colon \tilde{X} \to \tilde{Y} making the following diagram commute"
to
"there exits a continuous  h\colon \tilde{X} \to \tilde{Y} making the following diagram commute"

Without specifying base points, there are many continuous maps making the diagram commute.

page 291 line 19

 \theta\colon H \to G//H
to
 \theta\colon X \to G//H

page 297 line 11

 f_{t} = f_{t_{0}} * \lambda
to
 f_{t} \sim f_{t_{0}} * \lambda

page 300 line -8

The sentence "Replacing  V by  V \cap U if necessary, we may assume that  V \subset U." is verbose.
Since,  F(V \times I) \subset U implies  V = F(V \times \{0\}) \subset U.

page 302 line 12

"It is easy to see that  \tilde{X}_{G} is a group with identity  \tilde{e} and with  [\bar{f}] the inverse of  [f]"
to
"It is easy to see that  \tilde{X}_{G} is a group with identity  \tilde{e} and with  {\langle \bar{f} \rangle}_{G} the inverse of  {\langle f \rangle}_{G}"

page 308 line -4

"if  U is open in  X"
to
"if  U is open in  \tilde{X}/H"

page 313 line -12

 x \in U \subset \bar{U} \subset W
to
 y \in U \subset \bar{U} \subset W

page 318 line 10

Right parenthesis for  (Hint\colon is missing.

page 323 line -15

"there are no cogroups in  \textbf{hTop}"
to
"there are no non-trivial cogroups in  \textbf{hTop}"

The empty set  \emptyset is the trivial cogroup object in  \textbf{hTop}.

page 324 line -3

In Lemma 11.6(i),  Z is used to represent a space, but  X is used in the proof.

page 331

I think the proof of the Theorem 11.12 is too concise or rather imprecise.

Suppose  F\colon \Sigma X \to Y is a continuous map.
If we define  \bar{F}\colon X \times I \to Y by  \bar{F} = F \circ \nu where  \nu\colon X \times I \to \Sigma X is a natural map,
then we can define  \tau_{XY} by  [F \to [\bar{F}^{\#}]].

In addition, if  H\colon \Sigma X \times I \to Y is a (pointed) homotopy from  F_{0} to  F_{1}, association of  H
is not a (pointed) homotopy from  F_{0}^{\#} to  F_{1}^{\#}.

So, first, we define  \bar{H} by  H \circ (\nu \times I) \circ u\colon X \times I \times I \to Y where
 \nu is a natural map and u is a same map as in Theorem 11.8.
Then we can show that  \bar{H}^{\#} is a (pointed) homotopy from  \bar{F}_{0}^{\#} to  \bar{F}_{1}^{\#}.

Similary, suppose  G\colon X \to \Omega Y is a continuous map,
If we define  \tilde{G^{\flat}}\colon \Sigma X \to Y by  \tilde{G^{\flat}} = G^{\flat} \circ \nu^{-1},
then we can define  {\tau_{XY}}^{-1} by  [G] to  [\tilde{G^{\flat}}].

If  H\colon X \times I \to \Omega Y is a (pointed) homotopy from  G_{0} to  G_{1},
then we can show  H^{\flat} \circ u \circ (\nu \times 1)^{-1} is a (pointed) homotopy from  \tilde{G_{0}^{\flat}} to  \tilde{G_{1}^{\flat}}.


I guess that it is hard to read this contents from the proof.

page 332 line 5

for all  \omega, \omega' \in \Omega X
to
for all  \omega, \omega' \in \Omega Y

page 334 line 9

 \Sigma S^{n} = S^{n} \wedge S^{1} = (\textbf{R}^{n})^{\infty} \wedge \textbf{R}^{\infty} = (\textbf{R}^{n} \times \textbf{R})^{\infty} = (\textbf{R}^{n+1})^{\infty} = S^{n+1}
to
 \Sigma S^{n} \approx S^{n} \wedge S^{1} \approx (\textbf{R}^{n})^{\infty} \wedge \textbf{R}^{\infty} \approx (\textbf{R}^{n} \times \textbf{R})^{\infty} = (\textbf{R}^{n+1})^{\infty} \approx S^{n+1}

page 336 line -4

"Recall the identities  \dot{\textbf{I}}^{n} = (\dot{\textbf{I}}^{n-1} \times \textbf{I}) \cup (\textbf{I}^{n-1} \times \dot{\textbf{I}})
and  (\textbf{I}^{n-1} \times \textbf{I})/(\dot{\textbf{I}}^{n-1}  \times \textbf{I}) = (\textbf{I}^{n-1}/\dot{\textbf{I}}^{n-1}) \times \textbf{I}"

I guess the second identity does not hold.
When  n = 2,  (\textbf{I} \times \textbf{I})/(\dot{\textbf{I}} \times \textbf{I}) means that
for all  s,  (0,s) and  (1,s) are identified to a single point.
On the other hand,  (\textbf{I}/\dot{\textbf{I}}) \times \textbf{I} means that for each  s,  (0,s) and  (1,s) is identified to a point,
but, for distinct  s, s',  [0,s] and  [0,s'] are still distinct.

page 339 line -13

" \textit{Let}  \alpha, \beta \in \mathcal{Q}(X,x_{0})."

In Lemma 11.23(i)  \beta \in \mathcal{Q}(X,x_{0}), but, in Lemma 11,23(ii),  \beta \in \mathcal{Q}(X,x_{1}).

page 347 line 17

 (x_{0}, \omega_{0}, \beta)
to
 (x_{0}, \omega_{0}, \beta^{-1})

page 348 line 7

 \omega_{0} \simeq f
to
 c \simeq f, where  c is the constant map at  y_{0}.

page 338 line -10

 F\colon fg \simeq c rel  x_{0}, where  c is the constant map at  x_{0}.
to
 F\colon c \simeq fg rel  z_{0}, where  z_{0} is the base point of  Z and  c is the constant map at  y_{0}.

The order of the homolopy  F is sensitive, since it is used to define  \phi\colon Z \to M(fg).

page 353 line -7

"Let  \beta\colon (D^{n+1},S^{n}) \to (X, A) be a pointed map, and assume further that  \beta(0) = x_{0}"

For  \tilde{\beta} to be a pointed map,  \tilde{\beta}(s_{n}) = (x_{0}, \omega_{s_{n}}) must be a base point of  Mi,
namely,  \omega_{s_{n}} must be a constant map at  x_{0}.
So, the condition  \beta(0) = x_{0} is not enough.
We need that for all  t \in \textbf{I},  \omega_{s_{n}}(t) = \beta(ts_{n}) = x_{0}.

If  \gamma(D^{n+1},S^{n}) \to (X,A) is any pointed map, there is a pointed pair homotopy  \gamma \simeq \gamma \xi,
and  \gamma \xi(ts_{n}) = x_{0}, since explicit formula of Lemma 11.41 is  F(z,s) = z + (1 - |z|)ss_{n}, and  F(ts_{n},1) = s_{n}.

This further condition does hold.

page 363 line 2

"Note that  h is well defined (i.e.,  h(u,0) = x_{0} for all  u \in S^{n-1})"

Why does this hold?
From the definition of  h on  D^{n} \times \{0\}, forall  u \in S^{n-1},  h(u, 0) = f(u) \in F, and
generally  f(u) \neq x_{0}.

I guess it works fine if we define  h only on  D^{n} \times \{0\}.

page 365 line 4

 G(\{u\} \times [0, b_{1}])
to
 G(\{u\} \times [0, t_{1}])

page 365 line 7

 \tilde{h}_{0}\colon L^{(0)} \times [0,t] \to E
to
 \tilde{h}_{0}\colon L^{(0)} \times [0,t_{1}] \to E

page 365 line 14

 \tilde{\nu}_{\sigma}|\sigma \times \{0\} = \tilde{f}i|\sigma \times \{0\}
to
 \tilde{\nu}_{\sigma}|\sigma \times \{0\} = \tilde{f}i^{-1}|\sigma \times \{0\}

page 365 line -8

 h\sigma(u, t)
to
 h^{\sigma}(u,t)

page 380 line 4

 P = \{P_{n}\colon S_{n}(X) \to S_{n+1}(Y)\}
to
 P = \{P_{n}\colon S_{n}(X) \to S_{n+1}(X \times \textbf{I})\}

page 382 line -6

"(Hint \colon If the projection  \sum A_{\lambda} \to A_{\lambda} is denoted by  p_{\lambda} and
if  f \colon \sum A_{\lambda} \to G, then  f \mapsto (p_{\lambda}f) is an isomorphism.)"
to
"(Hint \colon If the injection  A_{\lambda} \to \sum A_{\lambda} is denoted by  j_{\lambda} and
if  f \colon \sum A_{\lambda} \to G, then  f \mapsto (fj_{\lambda}) is an isomorphism.)"

page 387 line -8

 F_{n}/F'_{n-1}
to
 F_{n}/F'_{n}

page 387 line -3

 \alpha' \in F'_{n}
to
 \alpha' \in F'_{n-1}

page 403 line 4

"It can be shown that  R \otimes S is the coproduct of  R and  S in the category of rings(or the category of graded ring)"
to
"It can be shown that  R \otimes S is the coproduct of  R and  S in the category of  \textbf{commutative} rings
(or the category of  \textbf{commutative} graded ring)"

参考文献

An Introduction to Algebraic Topology (Graduate Texts in Mathematics)

An Introduction to Algebraic Topology (Graduate Texts in Mathematics)