Every group the homology of some space?The homology group of the projective space of dimension $2$Products of homology groupsTopological Groups and the Mapping Class GroupHomology group of space $X$ given by the labelling scheme $aabcb^-1c^-1$Homology of a co-h-space manifoldproblem regarding fundamental and homology groupsQuestions about complexes and homologyProjective space, explicit descriptions of isomorphism between homology.Homology with real coefficientsHow to compute (co)homology group of the Eilenberg-Maclane space $K(pi,1)$
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Every group the homology of some space?
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Every group the homology of some space?
The homology group of the projective space of dimension $2$Products of homology groupsTopological Groups and the Mapping Class GroupHomology group of space $X$ given by the labelling scheme $aabcb^-1c^-1$Homology of a co-h-space manifoldproblem regarding fundamental and homology groupsQuestions about complexes and homologyProjective space, explicit descriptions of isomorphism between homology.Homology with real coefficientsHow to compute (co)homology group of the Eilenberg-Maclane space $K(pi,1)$
$begingroup$
Given a specific group, up to isomorphism, is there a way to determine a topological space, up to homeomorphism, with said group as the nth homology?
In other words, is there an established algorithm to work backwards from a specific group (like $mathbbZ_2$) and end up with some topological space?
abstract-algebra general-topology algebraic-topology category-theory
asked 1 hour ago
Jacob ClevelandJacob Cleveland
156
New contributor
Jacob Cleveland is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$
add a comment |
$begingroup$
Given a specific group, up to isomorphism, is there a way to determine a topological space, up to homeomorphism, with said group as the nth homology?
In other words, is there an established algorithm to work backwards from a specific group (like $mathbbZ_2$) and end up with some topological space?
abstract-algebra general-topology algebraic-topology category-theory
asked 1 hour ago
Jacob ClevelandJacob Cleveland
156
New contributor
Jacob Cleveland is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Given a specific group, up to isomorphism, is there a way to determine a topological space, up to homeomorphism, with said group as the nth homology?
In other words, is there an established algorithm to work backwards from a specific group (like $mathbbZ_2$) and end up with some topological space?
abstract-algebra general-topology algebraic-topology category-theory
asked 1 hour ago
Jacob ClevelandJacob Cleveland
156
New contributor
Jacob Cleveland is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Given a specific group, up to isomorphism, is there a way to determine a topological space, up to homeomorphism, with said group as the nth homology?
In other words, is there an established algorithm to work backwards from a specific group (like $mathbbZ_2$) and end up with some topological space?
abstract-algebra general-topology algebraic-topology category-theory
abstract-algebra general-topology algebraic-topology category-theory
asked 1 hour ago
Jacob ClevelandJacob Cleveland
156
New contributor
Jacob Cleveland is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
Jacob ClevelandJacob Cleveland
156
New contributor
Jacob Cleveland is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
Jacob ClevelandJacob Cleveland
156
New contributor
Jacob Cleveland is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
Jacob ClevelandJacob Cleveland
156
asked 1 hour ago
Jacob ClevelandJacob Cleveland
156
156
New contributor
Jacob Cleveland is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Jacob Cleveland is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
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$begingroup$
The only requirement is that the group is abelian. One such construction is the Eilenberg-MacLane space which is a space such that all homotopy groups except one are trivial. For existence see chapter 4 of Hatcher. These spaces answer your question because the Hurewicz Theorem tells us that the nth homology will be the nth homotopy group since the space is n-1 connected.
Another construction is the Moore Space which is the the homology analogue of the Eilenberg-MacLane space, also constructed in Hatcher.
answered 1 hour ago
Connor MalinConnor Malin
991112
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$begingroup$
The only requirement is that the group is abelian. One such construction is the Eilenberg-MacLane space which is a space such that all homotopy groups except one are trivial. For existence see chapter 4 of Hatcher. These spaces answer your question because the Hurewicz Theorem tells us that the nth homology will be the nth homotopy group since the space is n-1 connected.
Another construction is the Moore Space which is the the homology analogue of the Eilenberg-MacLane space, also constructed in Hatcher.
answered 1 hour ago
Connor MalinConnor Malin
991112
$endgroup$
add a comment |
$begingroup$
The only requirement is that the group is abelian. One such construction is the Eilenberg-MacLane space which is a space such that all homotopy groups except one are trivial. For existence see chapter 4 of Hatcher. These spaces answer your question because the Hurewicz Theorem tells us that the nth homology will be the nth homotopy group since the space is n-1 connected.
Another construction is the Moore Space which is the the homology analogue of the Eilenberg-MacLane space, also constructed in Hatcher.
answered 1 hour ago
Connor MalinConnor Malin
991112
$endgroup$
add a comment |
$begingroup$
The only requirement is that the group is abelian. One such construction is the Eilenberg-MacLane space which is a space such that all homotopy groups except one are trivial. For existence see chapter 4 of Hatcher. These spaces answer your question because the Hurewicz Theorem tells us that the nth homology will be the nth homotopy group since the space is n-1 connected.
Another construction is the Moore Space which is the the homology analogue of the Eilenberg-MacLane space, also constructed in Hatcher.
answered 1 hour ago
Connor MalinConnor Malin
991112
$endgroup$
The only requirement is that the group is abelian. One such construction is the Eilenberg-MacLane space which is a space such that all homotopy groups except one are trivial. For existence see chapter 4 of Hatcher. These spaces answer your question because the Hurewicz Theorem tells us that the nth homology will be the nth homotopy group since the space is n-1 connected.
Another construction is the Moore Space which is the the homology analogue of the Eilenberg-MacLane space, also constructed in Hatcher.
answered 1 hour ago
Connor MalinConnor Malin
991112
edited 1 hour ago
answered 1 hour ago
Connor MalinConnor Malin
991112
answered 1 hour ago
Connor MalinConnor Malin
991112
answered 1 hour ago
Connor MalinConnor Malin
991112
991112
add a comment |
add a comment |
Jacob Cleveland is a new contributor. Be nice, and check out our Code of Conduct.
Jacob Cleveland is a new contributor. Be nice, and check out our Code of Conduct.
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