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...
416213
$endgroup$
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...
416213
$endgroup$
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...
416213
$endgroup$
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...
416213
asked Dec 9 '18 at 14:38
123...123...
416213
edited Dec 9 '18 at 14:47
123...
asked Dec 9 '18 at 14:38
123...123...
416213
asked Dec 9 '18 at 14:38
123...123...
416213
asked Dec 9 '18 at 14:38
123...123...
416213
416213
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 |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3032433%2ffree-topological-g-space-and-g-cw-complex%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3032433%2ffree-topological-g-space-and-g-cw-complex%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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