Removing Homology Groups
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I was trying to construct a space that has first $n$ homology groups any given abelian groups $G_1, ..., G_n$. To show this I would like to be able to do the following: Given any space $X$, I can form some $X'$ such that $H_j(X') = 0$ some fixed $j$ and $H_i (X) = H_i (X')$ for all $i not = j$, i.e. a process of 'filling in $j$-dimensional holes'.
I do not see a way to proceed. It seems plausible but perhaps as I can only draw 'nice' spaces in my head.
Does any such process exist?
algebraic-topology homology-cohomology
asked Jan 5 at 19:18
IsomorphismIsomorphism
1198
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add a comment |
$begingroup$
I was trying to construct a space that has first $n$ homology groups any given abelian groups $G_1, ..., G_n$. To show this I would like to be able to do the following: Given any space $X$, I can form some $X'$ such that $H_j(X') = 0$ some fixed $j$ and $H_i (X) = H_i (X')$ for all $i not = j$, i.e. a process of 'filling in $j$-dimensional holes'.
I do not see a way to proceed. It seems plausible but perhaps as I can only draw 'nice' spaces in my head.
Does any such process exist?
algebraic-topology homology-cohomology
asked Jan 5 at 19:18
IsomorphismIsomorphism
1198
$endgroup$
add a comment |
$begingroup$
I was trying to construct a space that has first $n$ homology groups any given abelian groups $G_1, ..., G_n$. To show this I would like to be able to do the following: Given any space $X$, I can form some $X'$ such that $H_j(X') = 0$ some fixed $j$ and $H_i (X) = H_i (X')$ for all $i not = j$, i.e. a process of 'filling in $j$-dimensional holes'.
I do not see a way to proceed. It seems plausible but perhaps as I can only draw 'nice' spaces in my head.
Does any such process exist?
algebraic-topology homology-cohomology
asked Jan 5 at 19:18
IsomorphismIsomorphism
1198
$endgroup$
I was trying to construct a space that has first $n$ homology groups any given abelian groups $G_1, ..., G_n$. To show this I would like to be able to do the following: Given any space $X$, I can form some $X'$ such that $H_j(X') = 0$ some fixed $j$ and $H_i (X) = H_i (X')$ for all $i not = j$, i.e. a process of 'filling in $j$-dimensional holes'.
I do not see a way to proceed. It seems plausible but perhaps as I can only draw 'nice' spaces in my head.
Does any such process exist?
algebraic-topology homology-cohomology
algebraic-topology homology-cohomology
asked Jan 5 at 19:18
IsomorphismIsomorphism
1198
asked Jan 5 at 19:18
IsomorphismIsomorphism
1198
asked Jan 5 at 19:18
IsomorphismIsomorphism
1198
asked Jan 5 at 19:18
IsomorphismIsomorphism
1198
asked Jan 5 at 19:18
IsomorphismIsomorphism
1198
1198
add a comment |
add a comment |
1 Answer
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You can do this by taking a wedge product of Moore spaces. Your question can then be answered by reading how Moore spaces are constructed. See Hatcher's book, Example 2.40 and Example 2.41 (2015 print).
answered Jan 5 at 19:20
Pedro Tamaroff♦Pedro Tamaroff
96.4k10152297
$endgroup$
$begingroup$
I see that this allows us to solve my original question, but out of interest is it still possible to remove homology groups in general?
$endgroup$
– Isomorphism
Jan 5 at 19:27
$begingroup$
@Isomorphism The ideas used to construct Moore spaces work equally well to achieve your goals, but at the moment I'd have to go through them in a bit more detail to give you a more instructive answer. Now, I can tell you that the main tool to use is the cellular complex of your space (which you can assume is a CW-complex). Adding cells with appropriate characteristic maps will change the differential accordingly to kill the cycles you want.
$endgroup$
– Pedro Tamaroff♦
Jan 5 at 19:50
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You can do this by taking a wedge product of Moore spaces. Your question can then be answered by reading how Moore spaces are constructed. See Hatcher's book, Example 2.40 and Example 2.41 (2015 print).
answered Jan 5 at 19:20
Pedro Tamaroff♦Pedro Tamaroff
96.4k10152297
$endgroup$
$begingroup$
I see that this allows us to solve my original question, but out of interest is it still possible to remove homology groups in general?
$endgroup$
– Isomorphism
Jan 5 at 19:27
$begingroup$
@Isomorphism The ideas used to construct Moore spaces work equally well to achieve your goals, but at the moment I'd have to go through them in a bit more detail to give you a more instructive answer. Now, I can tell you that the main tool to use is the cellular complex of your space (which you can assume is a CW-complex). Adding cells with appropriate characteristic maps will change the differential accordingly to kill the cycles you want.
$endgroup$
– Pedro Tamaroff♦
Jan 5 at 19:50
add a comment |
$begingroup$
You can do this by taking a wedge product of Moore spaces. Your question can then be answered by reading how Moore spaces are constructed. See Hatcher's book, Example 2.40 and Example 2.41 (2015 print).
answered Jan 5 at 19:20
Pedro Tamaroff♦Pedro Tamaroff
96.4k10152297
$endgroup$
$begingroup$
I see that this allows us to solve my original question, but out of interest is it still possible to remove homology groups in general?
$endgroup$
– Isomorphism
Jan 5 at 19:27
$begingroup$
@Isomorphism The ideas used to construct Moore spaces work equally well to achieve your goals, but at the moment I'd have to go through them in a bit more detail to give you a more instructive answer. Now, I can tell you that the main tool to use is the cellular complex of your space (which you can assume is a CW-complex). Adding cells with appropriate characteristic maps will change the differential accordingly to kill the cycles you want.
$endgroup$
– Pedro Tamaroff♦
Jan 5 at 19:50
add a comment |
$begingroup$
You can do this by taking a wedge product of Moore spaces. Your question can then be answered by reading how Moore spaces are constructed. See Hatcher's book, Example 2.40 and Example 2.41 (2015 print).
answered Jan 5 at 19:20
Pedro Tamaroff♦Pedro Tamaroff
96.4k10152297
$endgroup$
You can do this by taking a wedge product of Moore spaces. Your question can then be answered by reading how Moore spaces are constructed. See Hatcher's book, Example 2.40 and Example 2.41 (2015 print).
answered Jan 5 at 19:20
Pedro Tamaroff♦Pedro Tamaroff
96.4k10152297
answered Jan 5 at 19:20
Pedro Tamaroff♦Pedro Tamaroff
96.4k10152297
answered Jan 5 at 19:20
Pedro Tamaroff♦Pedro Tamaroff
96.4k10152297
answered Jan 5 at 19:20
Pedro Tamaroff♦Pedro Tamaroff
96.4k10152297
96.4k10152297
$begingroup$
I see that this allows us to solve my original question, but out of interest is it still possible to remove homology groups in general?
$endgroup$
– Isomorphism
Jan 5 at 19:27
$begingroup$
@Isomorphism The ideas used to construct Moore spaces work equally well to achieve your goals, but at the moment I'd have to go through them in a bit more detail to give you a more instructive answer. Now, I can tell you that the main tool to use is the cellular complex of your space (which you can assume is a CW-complex). Adding cells with appropriate characteristic maps will change the differential accordingly to kill the cycles you want.
$endgroup$
– Pedro Tamaroff♦
Jan 5 at 19:50
add a comment |
$begingroup$
I see that this allows us to solve my original question, but out of interest is it still possible to remove homology groups in general?
$endgroup$
– Isomorphism
Jan 5 at 19:27
$begingroup$
@Isomorphism The ideas used to construct Moore spaces work equally well to achieve your goals, but at the moment I'd have to go through them in a bit more detail to give you a more instructive answer. Now, I can tell you that the main tool to use is the cellular complex of your space (which you can assume is a CW-complex). Adding cells with appropriate characteristic maps will change the differential accordingly to kill the cycles you want.
$endgroup$
– Pedro Tamaroff♦
Jan 5 at 19:50
$begingroup$
I see that this allows us to solve my original question, but out of interest is it still possible to remove homology groups in general?
$endgroup$
– Isomorphism
Jan 5 at 19:27
$begingroup$
I see that this allows us to solve my original question, but out of interest is it still possible to remove homology groups in general?
$endgroup$
– Isomorphism
Jan 5 at 19:27
$begingroup$
@Isomorphism The ideas used to construct Moore spaces work equally well to achieve your goals, but at the moment I'd have to go through them in a bit more detail to give you a more instructive answer. Now, I can tell you that the main tool to use is the cellular complex of your space (which you can assume is a CW-complex). Adding cells with appropriate characteristic maps will change the differential accordingly to kill the cycles you want.
$endgroup$
– Pedro Tamaroff♦
Jan 5 at 19:50
$begingroup$
@Isomorphism The ideas used to construct Moore spaces work equally well to achieve your goals, but at the moment I'd have to go through them in a bit more detail to give you a more instructive answer. Now, I can tell you that the main tool to use is the cellular complex of your space (which you can assume is a CW-complex). Adding cells with appropriate characteristic maps will change the differential accordingly to kill the cycles you want.
$endgroup$
– Pedro Tamaroff♦
Jan 5 at 19:50
add a comment |
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