Free topological $G$-space and G-CW-Complex
$begingroup$
Let $G$ be a finite group, and $X$ a free topological $G$-space which admits a CW-structure. Is there a CW-structure on $X$ that compatible with its $G$-action, i.e, a cell structure that turns $X$ to a G-CW-complex?
Note, the n-sphere $mathbb{S}^n$ with the antipodal action, i.e., $xto -x$ is an example of free $mathbb{Z}_2$-space. The cell structure of $mathbb{S}^n$ with just two cells does not respect the action while the usual cell structure which has two cells in each dimension does.
algebraic-topology
asked Dec 9 '18 at 14:38
123...123...
426213
$endgroup$
add a comment |
$begingroup$
Let $G$ be a finite group, and $X$ a free topological $G$-space which admits a CW-structure. Is there a CW-structure on $X$ that compatible with its $G$-action, i.e, a cell structure that turns $X$ to a G-CW-complex?
Note, the n-sphere $mathbb{S}^n$ with the antipodal action, i.e., $xto -x$ is an example of free $mathbb{Z}_2$-space. The cell structure of $mathbb{S}^n$ with just two cells does not respect the action while the usual cell structure which has two cells in each dimension does.
algebraic-topology
asked Dec 9 '18 at 14:38
123...123...
426213
$endgroup$
3
$begingroup$
Since $G$ is finite and acts freely, it is automatically a covering space action on $X$. Pass to the quotient $X/G$, give it a CW structure, and lift the CW structure cell-by-cell by the covering map $X to X/G$. This is a $G$-invariant CW structure on $X$.
$endgroup$
– Balarka Sen
Dec 9 '18 at 16:14
$begingroup$
Thanks for your time. Ok! let's do a practice to see how this observation work. For example, consider the sphere $S^{2n-1}subseteqmathbb{C}^n$ equipped with the $Z_m$-action induced by the map $(v_1,ldots, v_n)$ to $(e^{frac{2pi j}{m}}v_1,ldots, e^{frac{2pi j}{m}}v_n)$ where $jin Z_m$. Could you pleased describe a $mathbb{Z}_m$-cw-structure on $S^{2n-1}$ explicitly?
$endgroup$
– 123...
Dec 10 '18 at 13:31
add a comment |
$begingroup$
Let $G$ be a finite group, and $X$ a free topological $G$-space which admits a CW-structure. Is there a CW-structure on $X$ that compatible with its $G$-action, i.e, a cell structure that turns $X$ to a G-CW-complex?
Note, the n-sphere $mathbb{S}^n$ with the antipodal action, i.e., $xto -x$ is an example of free $mathbb{Z}_2$-space. The cell structure of $mathbb{S}^n$ with just two cells does not respect the action while the usual cell structure which has two cells in each dimension does.
algebraic-topology
asked Dec 9 '18 at 14:38
123...123...
426213
$endgroup$
Let $G$ be a finite group, and $X$ a free topological $G$-space which admits a CW-structure. Is there a CW-structure on $X$ that compatible with its $G$-action, i.e, a cell structure that turns $X$ to a G-CW-complex?
Note, the n-sphere $mathbb{S}^n$ with the antipodal action, i.e., $xto -x$ is an example of free $mathbb{Z}_2$-space. The cell structure of $mathbb{S}^n$ with just two cells does not respect the action while the usual cell structure which has two cells in each dimension does.
algebraic-topology
algebraic-topology
asked Dec 9 '18 at 14:38
123...123...
426213
asked Dec 9 '18 at 14:38
123...123...
426213
edited Dec 9 '18 at 14:47
123...
asked Dec 9 '18 at 14:38
123...123...
426213
asked Dec 9 '18 at 14:38
123...123...
426213
asked Dec 9 '18 at 14:38
123...123...
426213
426213
3
$begingroup$
Since $G$ is finite and acts freely, it is automatically a covering space action on $X$. Pass to the quotient $X/G$, give it a CW structure, and lift the CW structure cell-by-cell by the covering map $X to X/G$. This is a $G$-invariant CW structure on $X$.
$endgroup$
– Balarka Sen
Dec 9 '18 at 16:14
$begingroup$
Thanks for your time. Ok! let's do a practice to see how this observation work. For example, consider the sphere $S^{2n-1}subseteqmathbb{C}^n$ equipped with the $Z_m$-action induced by the map $(v_1,ldots, v_n)$ to $(e^{frac{2pi j}{m}}v_1,ldots, e^{frac{2pi j}{m}}v_n)$ where $jin Z_m$. Could you pleased describe a $mathbb{Z}_m$-cw-structure on $S^{2n-1}$ explicitly?
$endgroup$
– 123...
Dec 10 '18 at 13:31
add a comment |
3
$begingroup$
Since $G$ is finite and acts freely, it is automatically a covering space action on $X$. Pass to the quotient $X/G$, give it a CW structure, and lift the CW structure cell-by-cell by the covering map $X to X/G$. This is a $G$-invariant CW structure on $X$.
$endgroup$
– Balarka Sen
Dec 9 '18 at 16:14
$begingroup$
Thanks for your time. Ok! let's do a practice to see how this observation work. For example, consider the sphere $S^{2n-1}subseteqmathbb{C}^n$ equipped with the $Z_m$-action induced by the map $(v_1,ldots, v_n)$ to $(e^{frac{2pi j}{m}}v_1,ldots, e^{frac{2pi j}{m}}v_n)$ where $jin Z_m$. Could you pleased describe a $mathbb{Z}_m$-cw-structure on $S^{2n-1}$ explicitly?
$endgroup$
– 123...
Dec 10 '18 at 13:31
3
3
$begingroup$
Since $G$ is finite and acts freely, it is automatically a covering space action on $X$. Pass to the quotient $X/G$, give it a CW structure, and lift the CW structure cell-by-cell by the covering map $X to X/G$. This is a $G$-invariant CW structure on $X$.
$endgroup$
– Balarka Sen
Dec 9 '18 at 16:14
$begingroup$
Since $G$ is finite and acts freely, it is automatically a covering space action on $X$. Pass to the quotient $X/G$, give it a CW structure, and lift the CW structure cell-by-cell by the covering map $X to X/G$. This is a $G$-invariant CW structure on $X$.
$endgroup$
– Balarka Sen
Dec 9 '18 at 16:14
$begingroup$
Thanks for your time. Ok! let's do a practice to see how this observation work. For example, consider the sphere $S^{2n-1}subseteqmathbb{C}^n$ equipped with the $Z_m$-action induced by the map $(v_1,ldots, v_n)$ to $(e^{frac{2pi j}{m}}v_1,ldots, e^{frac{2pi j}{m}}v_n)$ where $jin Z_m$. Could you pleased describe a $mathbb{Z}_m$-cw-structure on $S^{2n-1}$ explicitly?
$endgroup$
– 123...
Dec 10 '18 at 13:31
$begingroup$
Thanks for your time. Ok! let's do a practice to see how this observation work. For example, consider the sphere $S^{2n-1}subseteqmathbb{C}^n$ equipped with the $Z_m$-action induced by the map $(v_1,ldots, v_n)$ to $(e^{frac{2pi j}{m}}v_1,ldots, e^{frac{2pi j}{m}}v_n)$ where $jin Z_m$. Could you pleased describe a $mathbb{Z}_m$-cw-structure on $S^{2n-1}$ explicitly?
$endgroup$
– 123...
Dec 10 '18 at 13:31
add a comment |
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$begingroup$
Since $G$ is finite and acts freely, it is automatically a covering space action on $X$. Pass to the quotient $X/G$, give it a CW structure, and lift the CW structure cell-by-cell by the covering map $X to X/G$. This is a $G$-invariant CW structure on $X$.
$endgroup$
– Balarka Sen
Dec 9 '18 at 16:14
$begingroup$
Thanks for your time. Ok! let's do a practice to see how this observation work. For example, consider the sphere $S^{2n-1}subseteqmathbb{C}^n$ equipped with the $Z_m$-action induced by the map $(v_1,ldots, v_n)$ to $(e^{frac{2pi j}{m}}v_1,ldots, e^{frac{2pi j}{m}}v_n)$ where $jin Z_m$. Could you pleased describe a $mathbb{Z}_m$-cw-structure on $S^{2n-1}$ explicitly?
$endgroup$
– 123...
Dec 10 '18 at 13:31