Homology with coefficient
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I have seen a rough statement in chapter 2 of Hatcher's Algebraic Toplogy book that states homology groups with coefficients in $mathbb{Z}$ contain less information rather than homology groups with coefficients in $mathbb{Z}_m$. Further, in a question from my professor, I asked why people use homology with integers coefficients and he said homology theory with coefficients in $mathbb{R}$ has less information. To which extent this statement holds? Is it true homology with coefficients in a bigger group contains less information?
algebraic-topology homology-cohomology
asked Jun 6 '17 at 7:09
mathvc_mathvc_
778
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add a comment |
$begingroup$
I have seen a rough statement in chapter 2 of Hatcher's Algebraic Toplogy book that states homology groups with coefficients in $mathbb{Z}$ contain less information rather than homology groups with coefficients in $mathbb{Z}_m$. Further, in a question from my professor, I asked why people use homology with integers coefficients and he said homology theory with coefficients in $mathbb{R}$ has less information. To which extent this statement holds? Is it true homology with coefficients in a bigger group contains less information?
algebraic-topology homology-cohomology
asked Jun 6 '17 at 7:09
mathvc_mathvc_
778
$endgroup$
add a comment |
$begingroup$
I have seen a rough statement in chapter 2 of Hatcher's Algebraic Toplogy book that states homology groups with coefficients in $mathbb{Z}$ contain less information rather than homology groups with coefficients in $mathbb{Z}_m$. Further, in a question from my professor, I asked why people use homology with integers coefficients and he said homology theory with coefficients in $mathbb{R}$ has less information. To which extent this statement holds? Is it true homology with coefficients in a bigger group contains less information?
algebraic-topology homology-cohomology
asked Jun 6 '17 at 7:09
mathvc_mathvc_
778
$endgroup$
I have seen a rough statement in chapter 2 of Hatcher's Algebraic Toplogy book that states homology groups with coefficients in $mathbb{Z}$ contain less information rather than homology groups with coefficients in $mathbb{Z}_m$. Further, in a question from my professor, I asked why people use homology with integers coefficients and he said homology theory with coefficients in $mathbb{R}$ has less information. To which extent this statement holds? Is it true homology with coefficients in a bigger group contains less information?
algebraic-topology homology-cohomology
algebraic-topology homology-cohomology
asked Jun 6 '17 at 7:09
mathvc_mathvc_
778
asked Jun 6 '17 at 7:09
mathvc_mathvc_
778
asked Jun 6 '17 at 7:09
mathvc_mathvc_
778
asked Jun 6 '17 at 7:09
mathvc_mathvc_
778
asked Jun 6 '17 at 7:09
mathvc_mathvc_
778
778
add a comment |
add a comment |
1 Answer
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homology groups with coefficients in $mathbb{Z}$ contain less information rather than homology groups with coefficients in $mathbb{Z}_m$
This is totally false. Homology with coefficients in $mathbb{Z}$ contains more information than homology with coefficients in any other group. Indeed, by the universal coefficient theorem, homology with any other coefficients is determined by homology with coefficients in $mathbb{Z}$.
The main reason we use other coefficients besides $mathbb{Z}$ despite them having less information is that they can be easier to work with. In general, working with coefficients in a field is easier, since then your homology groups are vector spaces (and in particular, they always have a basis). Another reason is that sometimes other coefficients arise naturally for geometric reasons. For instance, using coefficients in $mathbb{Z}_2$ gives us Poincaré duality even on nonorientable manifolds, and (co)homology of smooth manifolds with coefficients in $mathbb{R}$ is naturally related to differential forms by the de Rham theorem.
answered Jun 6 '17 at 7:17
Eric WofseyEric Wofsey
183k13211338
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Thank you very much for your answer.
$endgroup$
– mathvc_
Jun 6 '17 at 7:19
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
homology groups with coefficients in $mathbb{Z}$ contain less information rather than homology groups with coefficients in $mathbb{Z}_m$
This is totally false. Homology with coefficients in $mathbb{Z}$ contains more information than homology with coefficients in any other group. Indeed, by the universal coefficient theorem, homology with any other coefficients is determined by homology with coefficients in $mathbb{Z}$.
The main reason we use other coefficients besides $mathbb{Z}$ despite them having less information is that they can be easier to work with. In general, working with coefficients in a field is easier, since then your homology groups are vector spaces (and in particular, they always have a basis). Another reason is that sometimes other coefficients arise naturally for geometric reasons. For instance, using coefficients in $mathbb{Z}_2$ gives us Poincaré duality even on nonorientable manifolds, and (co)homology of smooth manifolds with coefficients in $mathbb{R}$ is naturally related to differential forms by the de Rham theorem.
answered Jun 6 '17 at 7:17
Eric WofseyEric Wofsey
183k13211338
$endgroup$
$begingroup$
Thank you very much for your answer.
$endgroup$
– mathvc_
Jun 6 '17 at 7:19
add a comment |
$begingroup$
homology groups with coefficients in $mathbb{Z}$ contain less information rather than homology groups with coefficients in $mathbb{Z}_m$
This is totally false. Homology with coefficients in $mathbb{Z}$ contains more information than homology with coefficients in any other group. Indeed, by the universal coefficient theorem, homology with any other coefficients is determined by homology with coefficients in $mathbb{Z}$.
The main reason we use other coefficients besides $mathbb{Z}$ despite them having less information is that they can be easier to work with. In general, working with coefficients in a field is easier, since then your homology groups are vector spaces (and in particular, they always have a basis). Another reason is that sometimes other coefficients arise naturally for geometric reasons. For instance, using coefficients in $mathbb{Z}_2$ gives us Poincaré duality even on nonorientable manifolds, and (co)homology of smooth manifolds with coefficients in $mathbb{R}$ is naturally related to differential forms by the de Rham theorem.
answered Jun 6 '17 at 7:17
Eric WofseyEric Wofsey
183k13211338
$endgroup$
$begingroup$
Thank you very much for your answer.
$endgroup$
– mathvc_
Jun 6 '17 at 7:19
add a comment |
$begingroup$
homology groups with coefficients in $mathbb{Z}$ contain less information rather than homology groups with coefficients in $mathbb{Z}_m$
This is totally false. Homology with coefficients in $mathbb{Z}$ contains more information than homology with coefficients in any other group. Indeed, by the universal coefficient theorem, homology with any other coefficients is determined by homology with coefficients in $mathbb{Z}$.
The main reason we use other coefficients besides $mathbb{Z}$ despite them having less information is that they can be easier to work with. In general, working with coefficients in a field is easier, since then your homology groups are vector spaces (and in particular, they always have a basis). Another reason is that sometimes other coefficients arise naturally for geometric reasons. For instance, using coefficients in $mathbb{Z}_2$ gives us Poincaré duality even on nonorientable manifolds, and (co)homology of smooth manifolds with coefficients in $mathbb{R}$ is naturally related to differential forms by the de Rham theorem.
answered Jun 6 '17 at 7:17
Eric WofseyEric Wofsey
183k13211338
$endgroup$
homology groups with coefficients in $mathbb{Z}$ contain less information rather than homology groups with coefficients in $mathbb{Z}_m$
This is totally false. Homology with coefficients in $mathbb{Z}$ contains more information than homology with coefficients in any other group. Indeed, by the universal coefficient theorem, homology with any other coefficients is determined by homology with coefficients in $mathbb{Z}$.
The main reason we use other coefficients besides $mathbb{Z}$ despite them having less information is that they can be easier to work with. In general, working with coefficients in a field is easier, since then your homology groups are vector spaces (and in particular, they always have a basis). Another reason is that sometimes other coefficients arise naturally for geometric reasons. For instance, using coefficients in $mathbb{Z}_2$ gives us Poincaré duality even on nonorientable manifolds, and (co)homology of smooth manifolds with coefficients in $mathbb{R}$ is naturally related to differential forms by the de Rham theorem.
answered Jun 6 '17 at 7:17
Eric WofseyEric Wofsey
183k13211338
edited Oct 7 '17 at 5:07
answered Jun 6 '17 at 7:17
Eric WofseyEric Wofsey
183k13211338
answered Jun 6 '17 at 7:17
Eric WofseyEric Wofsey
183k13211338
answered Jun 6 '17 at 7:17
Eric WofseyEric Wofsey
183k13211338
183k13211338
$begingroup$
Thank you very much for your answer.
$endgroup$
– mathvc_
Jun 6 '17 at 7:19
add a comment |
$begingroup$
Thank you very much for your answer.
$endgroup$
– mathvc_
Jun 6 '17 at 7:19
$begingroup$
Thank you very much for your answer.
$endgroup$
– mathvc_
Jun 6 '17 at 7:19
$begingroup$
Thank you very much for your answer.
$endgroup$
– mathvc_
Jun 6 '17 at 7:19
add a comment |
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