What are the attaching maps for the real Grassmannian?
$begingroup$
The Grassmannian $G_n(mathbb{R}^k)$ of n-planes in $mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition.
The study of characteristic classes tells us that these Schubert cells generate the cohomology of the Grassmannian and so the differentials in the cochain complex with $mathbb{Z}/2$ coefficients are all zero.
I am studying the $RO(mathbb{Z}/2)$-graded equivariant cohomology of Grassmann manifolds and there can be non-trivial differentials (in the appropriate spectral sequence) arising from a non-trivial attaching of a high dimensional cell to lower dimensional cells which will affect the cohomology. (This differs from the non-equivariant setting in that attaching an $n$-dimensional cell influences only the adjacent cohomology groups and no others. It turns out that attaching an $n$-dimensional cell can affect the equivariant cohomology in a larger range of dimensions.)
Hence, I am interested in knowing which of these Schubert cells are non-trivially attached to which other cells and how to detect this behavior.
general-topology algebraic-topology grassmannian schubert-calculus
edited Jan 5 at 11:19
Matt Samuel
39.2k63770
asked May 5 '11 at 16:05
wckronholmwckronholm
3,06711329
$endgroup$
add a comment |
$begingroup$
The Grassmannian $G_n(mathbb{R}^k)$ of n-planes in $mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition.
The study of characteristic classes tells us that these Schubert cells generate the cohomology of the Grassmannian and so the differentials in the cochain complex with $mathbb{Z}/2$ coefficients are all zero.
I am studying the $RO(mathbb{Z}/2)$-graded equivariant cohomology of Grassmann manifolds and there can be non-trivial differentials (in the appropriate spectral sequence) arising from a non-trivial attaching of a high dimensional cell to lower dimensional cells which will affect the cohomology. (This differs from the non-equivariant setting in that attaching an $n$-dimensional cell influences only the adjacent cohomology groups and no others. It turns out that attaching an $n$-dimensional cell can affect the equivariant cohomology in a larger range of dimensions.)
Hence, I am interested in knowing which of these Schubert cells are non-trivially attached to which other cells and how to detect this behavior.
general-topology algebraic-topology grassmannian schubert-calculus
edited Jan 5 at 11:19
Matt Samuel
39.2k63770
asked May 5 '11 at 16:05
wckronholmwckronholm
3,06711329
$endgroup$
add a comment |
$begingroup$
The Grassmannian $G_n(mathbb{R}^k)$ of n-planes in $mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition.
The study of characteristic classes tells us that these Schubert cells generate the cohomology of the Grassmannian and so the differentials in the cochain complex with $mathbb{Z}/2$ coefficients are all zero.
I am studying the $RO(mathbb{Z}/2)$-graded equivariant cohomology of Grassmann manifolds and there can be non-trivial differentials (in the appropriate spectral sequence) arising from a non-trivial attaching of a high dimensional cell to lower dimensional cells which will affect the cohomology. (This differs from the non-equivariant setting in that attaching an $n$-dimensional cell influences only the adjacent cohomology groups and no others. It turns out that attaching an $n$-dimensional cell can affect the equivariant cohomology in a larger range of dimensions.)
Hence, I am interested in knowing which of these Schubert cells are non-trivially attached to which other cells and how to detect this behavior.
general-topology algebraic-topology grassmannian schubert-calculus
edited Jan 5 at 11:19
Matt Samuel
39.2k63770
asked May 5 '11 at 16:05
wckronholmwckronholm
3,06711329
$endgroup$
The Grassmannian $G_n(mathbb{R}^k)$ of n-planes in $mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition.
The study of characteristic classes tells us that these Schubert cells generate the cohomology of the Grassmannian and so the differentials in the cochain complex with $mathbb{Z}/2$ coefficients are all zero.
I am studying the $RO(mathbb{Z}/2)$-graded equivariant cohomology of Grassmann manifolds and there can be non-trivial differentials (in the appropriate spectral sequence) arising from a non-trivial attaching of a high dimensional cell to lower dimensional cells which will affect the cohomology. (This differs from the non-equivariant setting in that attaching an $n$-dimensional cell influences only the adjacent cohomology groups and no others. It turns out that attaching an $n$-dimensional cell can affect the equivariant cohomology in a larger range of dimensions.)
Hence, I am interested in knowing which of these Schubert cells are non-trivially attached to which other cells and how to detect this behavior.
general-topology algebraic-topology grassmannian schubert-calculus
general-topology algebraic-topology grassmannian schubert-calculus
edited Jan 5 at 11:19
Matt Samuel
39.2k63770
asked May 5 '11 at 16:05
wckronholmwckronholm
3,06711329
edited Jan 5 at 11:19
Matt Samuel
39.2k63770
asked May 5 '11 at 16:05
wckronholmwckronholm
3,06711329
edited Jan 5 at 11:19
Matt Samuel
39.2k63770
edited Jan 5 at 11:19
Matt Samuel
39.2k63770
edited Jan 5 at 11:19
Matt Samuel
39.2k63770
39.2k63770
asked May 5 '11 at 16:05
wckronholmwckronholm
3,06711329
asked May 5 '11 at 16:05
wckronholmwckronholm
3,06711329
asked May 5 '11 at 16:05
wckronholmwckronholm
3,06711329
3,06711329
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This seems to be a difficult problem; in this link, it is mentioned that Bill Richter computed the attaching maps in his University of Washington PhD thesis using Morse theory (I haven't been able to find it online, but if you contact the university, they should be able to help). Also, they mention a "1934 Annals paper by Ehresmann" discussing this topic, and point out that this is before spectral sequences. There is an unanswered Mathoverflow question discussing all of this as well:https://mathoverflow.net/questions/19980/attaching-maps-for-grassmann-manifolds
edited Apr 13 '17 at 12:58
Community♦
1
answered Jun 11 '13 at 3:07
Brian RushtonBrian Rushton
11.5k85086
$endgroup$
add a comment |
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%2f37208%2fwhat-are-the-attaching-maps-for-the-real-grassmannian%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This seems to be a difficult problem; in this link, it is mentioned that Bill Richter computed the attaching maps in his University of Washington PhD thesis using Morse theory (I haven't been able to find it online, but if you contact the university, they should be able to help). Also, they mention a "1934 Annals paper by Ehresmann" discussing this topic, and point out that this is before spectral sequences. There is an unanswered Mathoverflow question discussing all of this as well:https://mathoverflow.net/questions/19980/attaching-maps-for-grassmann-manifolds
edited Apr 13 '17 at 12:58
Community♦
1
answered Jun 11 '13 at 3:07
Brian RushtonBrian Rushton
11.5k85086
$endgroup$
add a comment |
$begingroup$
This seems to be a difficult problem; in this link, it is mentioned that Bill Richter computed the attaching maps in his University of Washington PhD thesis using Morse theory (I haven't been able to find it online, but if you contact the university, they should be able to help). Also, they mention a "1934 Annals paper by Ehresmann" discussing this topic, and point out that this is before spectral sequences. There is an unanswered Mathoverflow question discussing all of this as well:https://mathoverflow.net/questions/19980/attaching-maps-for-grassmann-manifolds
edited Apr 13 '17 at 12:58
Community♦
1
answered Jun 11 '13 at 3:07
Brian RushtonBrian Rushton
11.5k85086
$endgroup$
add a comment |
$begingroup$
This seems to be a difficult problem; in this link, it is mentioned that Bill Richter computed the attaching maps in his University of Washington PhD thesis using Morse theory (I haven't been able to find it online, but if you contact the university, they should be able to help). Also, they mention a "1934 Annals paper by Ehresmann" discussing this topic, and point out that this is before spectral sequences. There is an unanswered Mathoverflow question discussing all of this as well:https://mathoverflow.net/questions/19980/attaching-maps-for-grassmann-manifolds
edited Apr 13 '17 at 12:58
Community♦
1
answered Jun 11 '13 at 3:07
Brian RushtonBrian Rushton
11.5k85086
$endgroup$
This seems to be a difficult problem; in this link, it is mentioned that Bill Richter computed the attaching maps in his University of Washington PhD thesis using Morse theory (I haven't been able to find it online, but if you contact the university, they should be able to help). Also, they mention a "1934 Annals paper by Ehresmann" discussing this topic, and point out that this is before spectral sequences. There is an unanswered Mathoverflow question discussing all of this as well:https://mathoverflow.net/questions/19980/attaching-maps-for-grassmann-manifolds
edited Apr 13 '17 at 12:58
Community♦
1
answered Jun 11 '13 at 3:07
Brian RushtonBrian Rushton
11.5k85086
edited Apr 13 '17 at 12:58
Community♦
1
edited Apr 13 '17 at 12:58
Community♦
1
edited Apr 13 '17 at 12:58
Community♦
1
1
answered Jun 11 '13 at 3:07
Brian RushtonBrian Rushton
11.5k85086
answered Jun 11 '13 at 3:07
Brian RushtonBrian Rushton
11.5k85086
answered Jun 11 '13 at 3:07
Brian RushtonBrian Rushton
11.5k85086
11.5k85086
add a comment |
add a comment |
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%2f37208%2fwhat-are-the-attaching-maps-for-the-real-grassmannian%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
