- はじめに
- Errata list
- page 53 line 5
- page 74 line -3
- page 77 line -7
- page 114 line 6
- page 118 line -4
- page 157 line 14
- page 159 line -12
- page 183 line 6
- page 189 line -6
- page 191 line -6
- page 213 line -13
- page 266 line -18
- page 276 line 14
- page 277 last line
- page 288 line 15
- page 291 line 19
- page 297 line 11
- page 300 line -8
- page 302 line 12
- page 308 line -4
- page 313 line -12
- page 318 line 10
- page 323 line -15
- page 324 line -3
- page 331
- page 332 line 5
- page 334 line 9
- page 336 line -4
- page 339 line -13
- page 347 line 17
- page 348 line 7
- page 338 line -10
- page 353 line -7
- page 363 line 2
- page 365 line 4
- page 365 line 7
- page 365 line 14
- page 365 line -8
- page 380 line 4
- page 382 line -6
- page 387 line -8
- page 387 line -3
- page 403 line 4
- 参考文献
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
to
page 74 line -3
to
equals to
, and
applying (or
) to
means that applying
(or
) to the equation
.
page 77 line -7
to
page 114 line 6
I think that the sentence "It is easy to see that both definitions of agree when X is convex."
should be deleted or be changed to
"It is easy to see that both definitions of agree when X is convex and
is affine."
(when is called
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 is not
.
page 118 line -4
to
page 157 line 14
"the first is 18 27 and the second is 27
9"
to
"the first is 27 18 and the second is 9
27"
page 159 line -12
to
page 183 line 6
to
page 189 line -6
to
page 191 line -6
left top corner of 2nd diagram
to
page 213 line -13
to
page 266 line -18
"Since is convex, we see that the model
is F-acyclic."
to
"Since is convex, we see that the model
is F-acyclic."
page 276 line 14
to
page 277 last line
"Consider the diagram of continuous maps"
to
"Consider the diagram of continuous maps"
page 288 line 15
"there exits a continuous
making the following diagram commute"
to
"there exits a continuous making the following diagram commute"
Without specifying base points, there are many continuous maps making the diagram commute.
page 291 line 19
to
page 297 line 11
to
page 300 line -8
The sentence "Replacing by
if necessary, we may assume that
." is verbose.
Since, implies
.
page 302 line 12
"It is easy to see that is a group with identity
and with
the inverse of
"
to
"It is easy to see that is a group with identity
and with
the inverse of
"
page 308 line -4
"if is open in
"
to
"if is open in
"
page 313 line -12
to
page 318 line 10
Right parenthesis for is missing.
page 323 line -15
"there are no cogroups in "
to
"there are no non-trivial cogroups in "
The empty set is the trivial cogroup object in
.
page 324 line -3
In Lemma 11.6(i), is used to represent a space, but
is used in the proof.
page 331
I think the proof of the Theorem 11.12 is too concise or rather imprecise.
Suppose is a continuous map.
If we define by
where
is a natural map,
then we can define by
\to [\bar{F}^{\#}]].
In addition, if is a (pointed) homotopy from
to
, association of
is not a (pointed) homotopy from to
.
So, first, we define by
where
is a natural map and u is a same map as in Theorem 11.8.
Then we can show that is a (pointed) homotopy from
to
.
Similary, suppose is a continuous map,
If we define by
,
then we can define by
] to
].
If is a (pointed) homotopy from
to
,
then we can show is a (pointed) homotopy from
to
.
I guess that it is hard to read this contents from the proof.
page 332 line 5
for all
to
for all
page 334 line 9
to
page 336 line -4
"Recall the identities
and "
I guess the second identity does not hold.
When ,
means that
for all ,
and
are identified to a single point.
On the other hand, means that for each
,
and
is identified to a point,
but, for distinct ,
,
and
are still distinct.
page 339 line -13
"
."
In Lemma 11.23(i) , but, in Lemma 11,23(ii),
.
page 347 line 17
to
page 348 line 7
to
, where
is the constant map at
.
page 338 line -10
rel
, where
is the constant map at
.
to
rel
, where
is the base point of
and
is the constant map at
.
The order of the homolopy is sensitive, since it is used to define
.
page 353 line -7
"Let be a pointed map, and assume further that
"
For to be a pointed map,
must be a base point of
,
namely, must be a constant map at
.
So, the condition is not enough.
We need that for all ,
.
If is any pointed map, there is a pointed pair homotopy
,
and , since explicit formula of Lemma 11.41 is
, and
.
This further condition does hold.
page 363 line 2
"Note that is well defined (i.e.,
for all
"
Why does this hold?
From the definition of on
, forall
,
, and
generally .
I guess it works fine if we define only on
.
page 365 line 4
to
page 365 line 7
to
page 365 line 14
to
page 365 line -8
to
page 380 line 4
to
page 382 line -6
"(Hint If the projection
is denoted by
and
if , then
is an isomorphism.)"
to
"(Hint If the injection
is denoted by
and
if , then
is an isomorphism.)"
page 387 line -8
to
page 387 line -3
to
page 403 line 4
"It can be shown that is the coproduct of
and
in the category of rings(or the category of graded ring)"
to
"It can be shown that is the coproduct of
and
in the category of
rings
(or the category of graded ring)"
参考文献

An Introduction to Algebraic Topology (Graduate Texts in Mathematics)
- 作者: Joseph Rotman
- 出版社/メーカー: Springer
- 発売日: 1998/07/01
- メディア: ハードカバー
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