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

はじめに

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

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page 74 line -3

to

equals to , and
applying (or ) to means that applying (or ) to the equation .

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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 .

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page 157 line 14

"the first is 18 27 and the second is 27 9"
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"the first is 27 18 and the second is 9 27"

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page 191 line -6

left top corner of 2nd diagram
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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."

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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.

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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 "

"if is open in "
to
"if is open in "

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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.

for all
to
for all

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), .

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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 .

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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.)"

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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)