Terminology about G- simplicial complexes
$begingroup$
For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $gin G$ stabilizes a given simplex $sigmasubseteq X$, then $g:sigmatosigma$ is in fact the identity map. This condition is nice because it implies that the quotient $X/G$ is glued out of simplices (rather than simplices modulo a finite group of permutations of their vertices).
Is there a standard term for this condition (or any related one) on the action $Gcurvearrowright X$?
at.algebraic-topology terminology group-actions simplicial-complexes equivariant
asked May 27 at 14:26
John PardonJohn Pardon
10.1k3 gold badges34 silver badges111 bronze badges
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$begingroup$
For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $gin G$ stabilizes a given simplex $sigmasubseteq X$, then $g:sigmatosigma$ is in fact the identity map. This condition is nice because it implies that the quotient $X/G$ is glued out of simplices (rather than simplices modulo a finite group of permutations of their vertices).
Is there a standard term for this condition (or any related one) on the action $Gcurvearrowright X$?
at.algebraic-topology terminology group-actions simplicial-complexes equivariant
asked May 27 at 14:26
John PardonJohn Pardon
10.1k3 gold badges34 silver badges111 bronze badges
$endgroup$
6
$begingroup$
This condition appears in several of my papers, and despite looking I have not found a standard term for it. One potentially useful way of thinking about it is that it is equivalent to saying that you can turn your simplicial complex into a semisimplicial set with a $G$-action.
$endgroup$
– Andy Putman
May 27 at 14:45
add a comment
|
$begingroup$
For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $gin G$ stabilizes a given simplex $sigmasubseteq X$, then $g:sigmatosigma$ is in fact the identity map. This condition is nice because it implies that the quotient $X/G$ is glued out of simplices (rather than simplices modulo a finite group of permutations of their vertices).
Is there a standard term for this condition (or any related one) on the action $Gcurvearrowright X$?
at.algebraic-topology terminology group-actions simplicial-complexes equivariant
asked May 27 at 14:26
John PardonJohn Pardon
10.1k3 gold badges34 silver badges111 bronze badges
$endgroup$
For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $gin G$ stabilizes a given simplex $sigmasubseteq X$, then $g:sigmatosigma$ is in fact the identity map. This condition is nice because it implies that the quotient $X/G$ is glued out of simplices (rather than simplices modulo a finite group of permutations of their vertices).
Is there a standard term for this condition (or any related one) on the action $Gcurvearrowright X$?
at.algebraic-topology terminology group-actions simplicial-complexes equivariant
at.algebraic-topology terminology group-actions simplicial-complexes equivariant
asked May 27 at 14:26
John PardonJohn Pardon
10.1k3 gold badges34 silver badges111 bronze badges
asked May 27 at 14:26
John PardonJohn Pardon
10.1k3 gold badges34 silver badges111 bronze badges
asked May 27 at 14:26
John PardonJohn Pardon
10.1k3 gold badges34 silver badges111 bronze badges
asked May 27 at 14:26
John PardonJohn Pardon
10.1k3 gold badges34 silver badges111 bronze badges
asked May 27 at 14:26
John PardonJohn Pardon
10.1k3 gold badges34 silver badges111 bronze badges
10.1k3 gold badges34 silver badges111 bronze badges
6
$begingroup$
This condition appears in several of my papers, and despite looking I have not found a standard term for it. One potentially useful way of thinking about it is that it is equivalent to saying that you can turn your simplicial complex into a semisimplicial set with a $G$-action.
$endgroup$
– Andy Putman
May 27 at 14:45
add a comment
|
6
$begingroup$
This condition appears in several of my papers, and despite looking I have not found a standard term for it. One potentially useful way of thinking about it is that it is equivalent to saying that you can turn your simplicial complex into a semisimplicial set with a $G$-action.
$endgroup$
– Andy Putman
May 27 at 14:45
6
6
$begingroup$
This condition appears in several of my papers, and despite looking I have not found a standard term for it. One potentially useful way of thinking about it is that it is equivalent to saying that you can turn your simplicial complex into a semisimplicial set with a $G$-action.
$endgroup$
– Andy Putman
May 27 at 14:45
$begingroup$
This condition appears in several of my papers, and despite looking I have not found a standard term for it. One potentially useful way of thinking about it is that it is equivalent to saying that you can turn your simplicial complex into a semisimplicial set with a $G$-action.
$endgroup$
– Andy Putman
May 27 at 14:45
add a comment
|
1 Answer
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I've never stumbled across a standard one in all of the literature, though I've definitely appreciated a remark by Ken Brown in his group cohomology bible: "The hypothesis that $G_sigma$ fixes $sigma$ pointwise is not very restrictive in practice. In the case of a simplicial action, for example, it can always be achieved by passage to the barycentric subdivision."
Some references have the pointwise-fixed assumption baked into the definition of a $G$-CW-complex, but not all. Ken Brown's book doesn't, and he calls a $G$-CW-complex "admissible" if we include the pointwise-fixed constraint. This doesn't seem to be standard though. (Now as for definite nonstandard terminology, Farb-Margalit's book "A Primer on Mapping Class Groups" calls it a group action "without rotations".)
answered May 27 at 16:01
Chris GerigChris Gerig
9,9012 gold badges53 silver badges94 bronze badges
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$begingroup$
My vote is to start using "admissible (simplicial) action" often.
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– Chris Gerig
Jun 2 at 18:33
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$begingroup$
I've never stumbled across a standard one in all of the literature, though I've definitely appreciated a remark by Ken Brown in his group cohomology bible: "The hypothesis that $G_sigma$ fixes $sigma$ pointwise is not very restrictive in practice. In the case of a simplicial action, for example, it can always be achieved by passage to the barycentric subdivision."
Some references have the pointwise-fixed assumption baked into the definition of a $G$-CW-complex, but not all. Ken Brown's book doesn't, and he calls a $G$-CW-complex "admissible" if we include the pointwise-fixed constraint. This doesn't seem to be standard though. (Now as for definite nonstandard terminology, Farb-Margalit's book "A Primer on Mapping Class Groups" calls it a group action "without rotations".)
answered May 27 at 16:01
Chris GerigChris Gerig
9,9012 gold badges53 silver badges94 bronze badges
$endgroup$
$begingroup$
My vote is to start using "admissible (simplicial) action" often.
$endgroup$
– Chris Gerig
Jun 2 at 18:33
add a comment
|
$begingroup$
I've never stumbled across a standard one in all of the literature, though I've definitely appreciated a remark by Ken Brown in his group cohomology bible: "The hypothesis that $G_sigma$ fixes $sigma$ pointwise is not very restrictive in practice. In the case of a simplicial action, for example, it can always be achieved by passage to the barycentric subdivision."
Some references have the pointwise-fixed assumption baked into the definition of a $G$-CW-complex, but not all. Ken Brown's book doesn't, and he calls a $G$-CW-complex "admissible" if we include the pointwise-fixed constraint. This doesn't seem to be standard though. (Now as for definite nonstandard terminology, Farb-Margalit's book "A Primer on Mapping Class Groups" calls it a group action "without rotations".)
answered May 27 at 16:01
Chris GerigChris Gerig
9,9012 gold badges53 silver badges94 bronze badges
$endgroup$
$begingroup$
My vote is to start using "admissible (simplicial) action" often.
$endgroup$
– Chris Gerig
Jun 2 at 18:33
add a comment
|
$begingroup$
I've never stumbled across a standard one in all of the literature, though I've definitely appreciated a remark by Ken Brown in his group cohomology bible: "The hypothesis that $G_sigma$ fixes $sigma$ pointwise is not very restrictive in practice. In the case of a simplicial action, for example, it can always be achieved by passage to the barycentric subdivision."
Some references have the pointwise-fixed assumption baked into the definition of a $G$-CW-complex, but not all. Ken Brown's book doesn't, and he calls a $G$-CW-complex "admissible" if we include the pointwise-fixed constraint. This doesn't seem to be standard though. (Now as for definite nonstandard terminology, Farb-Margalit's book "A Primer on Mapping Class Groups" calls it a group action "without rotations".)
answered May 27 at 16:01
Chris GerigChris Gerig
9,9012 gold badges53 silver badges94 bronze badges
$endgroup$
I've never stumbled across a standard one in all of the literature, though I've definitely appreciated a remark by Ken Brown in his group cohomology bible: "The hypothesis that $G_sigma$ fixes $sigma$ pointwise is not very restrictive in practice. In the case of a simplicial action, for example, it can always be achieved by passage to the barycentric subdivision."
Some references have the pointwise-fixed assumption baked into the definition of a $G$-CW-complex, but not all. Ken Brown's book doesn't, and he calls a $G$-CW-complex "admissible" if we include the pointwise-fixed constraint. This doesn't seem to be standard though. (Now as for definite nonstandard terminology, Farb-Margalit's book "A Primer on Mapping Class Groups" calls it a group action "without rotations".)
answered May 27 at 16:01
Chris GerigChris Gerig
9,9012 gold badges53 silver badges94 bronze badges
edited May 27 at 16:19
answered May 27 at 16:01
Chris GerigChris Gerig
9,9012 gold badges53 silver badges94 bronze badges
answered May 27 at 16:01
Chris GerigChris Gerig
9,9012 gold badges53 silver badges94 bronze badges
answered May 27 at 16:01
Chris GerigChris Gerig
9,9012 gold badges53 silver badges94 bronze badges
9,9012 gold badges53 silver badges94 bronze badges
$begingroup$
My vote is to start using "admissible (simplicial) action" often.
$endgroup$
– Chris Gerig
Jun 2 at 18:33
add a comment
|
$begingroup$
My vote is to start using "admissible (simplicial) action" often.
$endgroup$
– Chris Gerig
Jun 2 at 18:33
$begingroup$
My vote is to start using "admissible (simplicial) action" often.
$endgroup$
– Chris Gerig
Jun 2 at 18:33
$begingroup$
My vote is to start using "admissible (simplicial) action" often.
$endgroup$
– Chris Gerig
Jun 2 at 18:33
add a comment
|
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$begingroup$
This condition appears in several of my papers, and despite looking I have not found a standard term for it. One potentially useful way of thinking about it is that it is equivalent to saying that you can turn your simplicial complex into a semisimplicial set with a $G$-action.
$endgroup$
– Andy Putman
May 27 at 14:45