Contractibility of CW complex without Whitehead
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Suppose I have a CW complex $X$ with skeleta $(X_n)_{nge 0}$ such that $pi_k(X)=0$ for all $kge 0$. I want to conclude that $X$ is contractible without invoking Whitehead's theorem.
It would be enough to see that the identity $X$ is nullhomotopic. My strategy would be the following:
- Show that all inclusions $i_n:X_nhookrightarrow X$ are null-homotopic,
- Conclude somehow that the limit $varinjlim, (i_n)=mathrm{id}$ is null-homotopic.
homotopy-theory cw-complexes higher-homotopy-groups
asked Nov 17 at 21:16
FKranhold
1787
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0
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favorite
Suppose I have a CW complex $X$ with skeleta $(X_n)_{nge 0}$ such that $pi_k(X)=0$ for all $kge 0$. I want to conclude that $X$ is contractible without invoking Whitehead's theorem.
It would be enough to see that the identity $X$ is nullhomotopic. My strategy would be the following:
- Show that all inclusions $i_n:X_nhookrightarrow X$ are null-homotopic,
- Conclude somehow that the limit $varinjlim, (i_n)=mathrm{id}$ is null-homotopic.
homotopy-theory cw-complexes higher-homotopy-groups
asked Nov 17 at 21:16
FKranhold
1787
What is your question: Whether your strategy is adequate or do you want to see a proof?
– Paul Frost
Nov 17 at 23:10
I want to have a proof but I thought it is a good idea to tell you what I have so far. If there's a proof using another approach, I would be happy, too.
– FKranhold
Nov 18 at 10:50
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose I have a CW complex $X$ with skeleta $(X_n)_{nge 0}$ such that $pi_k(X)=0$ for all $kge 0$. I want to conclude that $X$ is contractible without invoking Whitehead's theorem.
It would be enough to see that the identity $X$ is nullhomotopic. My strategy would be the following:
- Show that all inclusions $i_n:X_nhookrightarrow X$ are null-homotopic,
- Conclude somehow that the limit $varinjlim, (i_n)=mathrm{id}$ is null-homotopic.
homotopy-theory cw-complexes higher-homotopy-groups
asked Nov 17 at 21:16
FKranhold
1787
Suppose I have a CW complex $X$ with skeleta $(X_n)_{nge 0}$ such that $pi_k(X)=0$ for all $kge 0$. I want to conclude that $X$ is contractible without invoking Whitehead's theorem.
It would be enough to see that the identity $X$ is nullhomotopic. My strategy would be the following:
- Show that all inclusions $i_n:X_nhookrightarrow X$ are null-homotopic,
- Conclude somehow that the limit $varinjlim, (i_n)=mathrm{id}$ is null-homotopic.
homotopy-theory cw-complexes higher-homotopy-groups
homotopy-theory cw-complexes higher-homotopy-groups
asked Nov 17 at 21:16
FKranhold
1787
asked Nov 17 at 21:16
FKranhold
1787
asked Nov 17 at 21:16
FKranhold
1787
asked Nov 17 at 21:16
FKranhold
1787
asked Nov 17 at 21:16
FKranhold
1787
1787
What is your question: Whether your strategy is adequate or do you want to see a proof?
– Paul Frost
Nov 17 at 23:10
I want to have a proof but I thought it is a good idea to tell you what I have so far. If there's a proof using another approach, I would be happy, too.
– FKranhold
Nov 18 at 10:50
add a comment |
What is your question: Whether your strategy is adequate or do you want to see a proof?
– Paul Frost
Nov 17 at 23:10
I want to have a proof but I thought it is a good idea to tell you what I have so far. If there's a proof using another approach, I would be happy, too.
– FKranhold
Nov 18 at 10:50
What is your question: Whether your strategy is adequate or do you want to see a proof?
– Paul Frost
Nov 17 at 23:10
What is your question: Whether your strategy is adequate or do you want to see a proof?
– Paul Frost
Nov 17 at 23:10
I want to have a proof but I thought it is a good idea to tell you what I have so far. If there's a proof using another approach, I would be happy, too.
– FKranhold
Nov 18 at 10:50
I want to have a proof but I thought it is a good idea to tell you what I have so far. If there's a proof using another approach, I would be happy, too.
– FKranhold
Nov 18 at 10:50
add a comment |
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What is your question: Whether your strategy is adequate or do you want to see a proof?
– Paul Frost
Nov 17 at 23:10
I want to have a proof but I thought it is a good idea to tell you what I have so far. If there's a proof using another approach, I would be happy, too.
– FKranhold
Nov 18 at 10:50