Mixing
Rationalization of topological groups and degree maps
$begingroup$
Suppose $G$ a finitely generated nilpotent topological group and we consider its rationalization $G_mathbb{Q}$. This space may fail to be a topological group, but it's always a group-like H-space.
We know that for rationalization of groups maps of the form $xto x^n$ are bijections. My question is: can we determine if $G$ is rational (as topological space) by the maps of the above form? For example is this theorem correct?
Theorem: A topological group $G$ is rational, that is $G$ is homotopy equivalent to $G_mathbb{Q}$, iff all the maps $xto x^n$ are homotopy equivalences.
at.algebraic-topology homotopy-theory topological-groups rational-homotopy-theory
asked Feb 2 at 13:28
Fat ninjaFat ninja
935
$endgroup$
add a comment |
$begingroup$
Suppose $G$ a finitely generated nilpotent topological group and we consider its rationalization $G_mathbb{Q}$. This space may fail to be a topological group, but it's always a group-like H-space.
We know that for rationalization of groups maps of the form $xto x^n$ are bijections. My question is: can we determine if $G$ is rational (as topological space) by the maps of the above form? For example is this theorem correct?
Theorem: A topological group $G$ is rational, that is $G$ is homotopy equivalent to $G_mathbb{Q}$, iff all the maps $xto x^n$ are homotopy equivalences.
at.algebraic-topology homotopy-theory topological-groups rational-homotopy-theory
asked Feb 2 at 13:28
Fat ninjaFat ninja
935
$endgroup$
add a comment |
$begingroup$
Suppose $G$ a finitely generated nilpotent topological group and we consider its rationalization $G_mathbb{Q}$. This space may fail to be a topological group, but it's always a group-like H-space.
We know that for rationalization of groups maps of the form $xto x^n$ are bijections. My question is: can we determine if $G$ is rational (as topological space) by the maps of the above form? For example is this theorem correct?
Theorem: A topological group $G$ is rational, that is $G$ is homotopy equivalent to $G_mathbb{Q}$, iff all the maps $xto x^n$ are homotopy equivalences.
at.algebraic-topology homotopy-theory topological-groups rational-homotopy-theory
asked Feb 2 at 13:28
Fat ninjaFat ninja
935
$endgroup$
Suppose $G$ a finitely generated nilpotent topological group and we consider its rationalization $G_mathbb{Q}$. This space may fail to be a topological group, but it's always a group-like H-space.
We know that for rationalization of groups maps of the form $xto x^n$ are bijections. My question is: can we determine if $G$ is rational (as topological space) by the maps of the above form? For example is this theorem correct?
Theorem: A topological group $G$ is rational, that is $G$ is homotopy equivalent to $G_mathbb{Q}$, iff all the maps $xto x^n$ are homotopy equivalences.
at.algebraic-topology homotopy-theory topological-groups rational-homotopy-theory
at.algebraic-topology homotopy-theory topological-groups rational-homotopy-theory
asked Feb 2 at 13:28
Fat ninjaFat ninja
935
asked Feb 2 at 13:28
Fat ninjaFat ninja
935
edited Feb 2 at 13:40
Fat ninja
asked Feb 2 at 13:28
Fat ninjaFat ninja
935
asked Feb 2 at 13:28
Fat ninjaFat ninja
935
asked Feb 2 at 13:28
Fat ninjaFat ninja
935
935
add a comment |
add a comment |
1 Answer
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$begingroup$
Yes, since the group power map induces multiplication by $n$ on homotopy groups, turning them into rational vector spaces.
answered Feb 2 at 16:15
Jeff StromJeff Strom
7,38522860
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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active
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votes
$begingroup$
Yes, since the group power map induces multiplication by $n$ on homotopy groups, turning them into rational vector spaces.
answered Feb 2 at 16:15
Jeff StromJeff Strom
7,38522860
$endgroup$
add a comment |
$begingroup$
Yes, since the group power map induces multiplication by $n$ on homotopy groups, turning them into rational vector spaces.
answered Feb 2 at 16:15
Jeff StromJeff Strom
7,38522860
$endgroup$
add a comment |
$begingroup$
Yes, since the group power map induces multiplication by $n$ on homotopy groups, turning them into rational vector spaces.
answered Feb 2 at 16:15
Jeff StromJeff Strom
7,38522860
$endgroup$
Yes, since the group power map induces multiplication by $n$ on homotopy groups, turning them into rational vector spaces.
answered Feb 2 at 16:15
Jeff StromJeff Strom
7,38522860
answered Feb 2 at 16:15
Jeff StromJeff Strom
7,38522860
answered Feb 2 at 16:15
Jeff StromJeff Strom
7,38522860
answered Feb 2 at 16:15
Jeff StromJeff Strom
7,38522860
7,38522860
add a comment |
add a comment |
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