Mixing
Exercise in Simplicial Homology
$begingroup$
In Basic Concepts of Algebraic Topology by Fred Croom, the homology groups of the $n$-skeleton of the closure of an $(n+1)$-simplex are computed in Theorem 2.9. (The geometric carrier of the complex is homeomorphic to the $n$-sphere).
Exercise 21 at the end of the chapter (Chapter 2) asks one to show that the
homology groups of positive dimension of the closure of a simplex are trivial and thus deduce that the homology groups of the $n$-sphere are trivial in dimensions $1,2,...,n-1$.
I'm not sure what the author has in mind for this exercise. Of course, the simplex is contractible (whereas the sphere is not), but the concept of homotopy has not yet been introduced. With the exception of the homology group of dimension equal to that of the simplex, it seems to me that solving the exercise amounts to repeating the computations in Theorem 2.9.
algebraic-topology homology-cohomology
edited Jan 25 at 11:42
fermesomme
5,18022351
asked Jan 23 at 16:04
John R. SkukalekJohn R. Skukalek
61
$endgroup$
add a comment |
$begingroup$
In Basic Concepts of Algebraic Topology by Fred Croom, the homology groups of the $n$-skeleton of the closure of an $(n+1)$-simplex are computed in Theorem 2.9. (The geometric carrier of the complex is homeomorphic to the $n$-sphere).
Exercise 21 at the end of the chapter (Chapter 2) asks one to show that the
homology groups of positive dimension of the closure of a simplex are trivial and thus deduce that the homology groups of the $n$-sphere are trivial in dimensions $1,2,...,n-1$.
I'm not sure what the author has in mind for this exercise. Of course, the simplex is contractible (whereas the sphere is not), but the concept of homotopy has not yet been introduced. With the exception of the homology group of dimension equal to that of the simplex, it seems to me that solving the exercise amounts to repeating the computations in Theorem 2.9.
algebraic-topology homology-cohomology
edited Jan 25 at 11:42
fermesomme
5,18022351
asked Jan 23 at 16:04
John R. SkukalekJohn R. Skukalek
61
$endgroup$
$begingroup$
What does "closure" of a simplex mean here?
$endgroup$
– fermesomme
Jan 25 at 11:41
$begingroup$
@fermesomme It means the complex that consists of all faces of the simplex.
$endgroup$
– John R. Skukalek
Jan 25 at 11:54
$begingroup$
It is a bit strange since the author hasn't introduced homology groups, so I'm not sure what he's asking. Probably to repeat the computations in the theorem. To calculate the singular homology groups of a sphere, there are a couple of different ways. One is to make use of exact sequences of $(S^n, D^{n+})$, another uses Mayer-Vietoris and induction.
$endgroup$
– fermesomme
Jan 25 at 11:55
$begingroup$
@fermesomme Only the homology groups of a simplicial complex have been introduced.
$endgroup$
– John R. Skukalek
Jan 25 at 15:59
add a comment |
$begingroup$
In Basic Concepts of Algebraic Topology by Fred Croom, the homology groups of the $n$-skeleton of the closure of an $(n+1)$-simplex are computed in Theorem 2.9. (The geometric carrier of the complex is homeomorphic to the $n$-sphere).
Exercise 21 at the end of the chapter (Chapter 2) asks one to show that the
homology groups of positive dimension of the closure of a simplex are trivial and thus deduce that the homology groups of the $n$-sphere are trivial in dimensions $1,2,...,n-1$.
I'm not sure what the author has in mind for this exercise. Of course, the simplex is contractible (whereas the sphere is not), but the concept of homotopy has not yet been introduced. With the exception of the homology group of dimension equal to that of the simplex, it seems to me that solving the exercise amounts to repeating the computations in Theorem 2.9.
algebraic-topology homology-cohomology
edited Jan 25 at 11:42
fermesomme
5,18022351
asked Jan 23 at 16:04
John R. SkukalekJohn R. Skukalek
61
$endgroup$
In Basic Concepts of Algebraic Topology by Fred Croom, the homology groups of the $n$-skeleton of the closure of an $(n+1)$-simplex are computed in Theorem 2.9. (The geometric carrier of the complex is homeomorphic to the $n$-sphere).
Exercise 21 at the end of the chapter (Chapter 2) asks one to show that the
homology groups of positive dimension of the closure of a simplex are trivial and thus deduce that the homology groups of the $n$-sphere are trivial in dimensions $1,2,...,n-1$.
I'm not sure what the author has in mind for this exercise. Of course, the simplex is contractible (whereas the sphere is not), but the concept of homotopy has not yet been introduced. With the exception of the homology group of dimension equal to that of the simplex, it seems to me that solving the exercise amounts to repeating the computations in Theorem 2.9.
algebraic-topology homology-cohomology
algebraic-topology homology-cohomology
edited Jan 25 at 11:42
fermesomme
5,18022351
asked Jan 23 at 16:04
John R. SkukalekJohn R. Skukalek
61
edited Jan 25 at 11:42
fermesomme
5,18022351
asked Jan 23 at 16:04
John R. SkukalekJohn R. Skukalek
61
edited Jan 25 at 11:42
fermesomme
5,18022351
edited Jan 25 at 11:42
fermesomme
5,18022351
edited Jan 25 at 11:42
fermesomme
5,18022351
5,18022351
asked Jan 23 at 16:04
John R. SkukalekJohn R. Skukalek
61
asked Jan 23 at 16:04
John R. SkukalekJohn R. Skukalek
61
asked Jan 23 at 16:04
John R. SkukalekJohn R. Skukalek
61
61
$begingroup$
What does "closure" of a simplex mean here?
$endgroup$
– fermesomme
Jan 25 at 11:41
$begingroup$
@fermesomme It means the complex that consists of all faces of the simplex.
$endgroup$
– John R. Skukalek
Jan 25 at 11:54
$begingroup$
It is a bit strange since the author hasn't introduced homology groups, so I'm not sure what he's asking. Probably to repeat the computations in the theorem. To calculate the singular homology groups of a sphere, there are a couple of different ways. One is to make use of exact sequences of $(S^n, D^{n+})$, another uses Mayer-Vietoris and induction.
$endgroup$
– fermesomme
Jan 25 at 11:55
$begingroup$
@fermesomme Only the homology groups of a simplicial complex have been introduced.
$endgroup$
– John R. Skukalek
Jan 25 at 15:59
add a comment |
$begingroup$
What does "closure" of a simplex mean here?
$endgroup$
– fermesomme
Jan 25 at 11:41
$begingroup$
@fermesomme It means the complex that consists of all faces of the simplex.
$endgroup$
– John R. Skukalek
Jan 25 at 11:54
$begingroup$
It is a bit strange since the author hasn't introduced homology groups, so I'm not sure what he's asking. Probably to repeat the computations in the theorem. To calculate the singular homology groups of a sphere, there are a couple of different ways. One is to make use of exact sequences of $(S^n, D^{n+})$, another uses Mayer-Vietoris and induction.
$endgroup$
– fermesomme
Jan 25 at 11:55
$begingroup$
@fermesomme Only the homology groups of a simplicial complex have been introduced.
$endgroup$
– John R. Skukalek
Jan 25 at 15:59
$begingroup$
What does "closure" of a simplex mean here?
$endgroup$
– fermesomme
Jan 25 at 11:41
$begingroup$
What does "closure" of a simplex mean here?
$endgroup$
– fermesomme
Jan 25 at 11:41
$begingroup$
@fermesomme It means the complex that consists of all faces of the simplex.
$endgroup$
– John R. Skukalek
Jan 25 at 11:54
$begingroup$
@fermesomme It means the complex that consists of all faces of the simplex.
$endgroup$
– John R. Skukalek
Jan 25 at 11:54
$begingroup$
It is a bit strange since the author hasn't introduced homology groups, so I'm not sure what he's asking. Probably to repeat the computations in the theorem. To calculate the singular homology groups of a sphere, there are a couple of different ways. One is to make use of exact sequences of $(S^n, D^{n+})$, another uses Mayer-Vietoris and induction.
$endgroup$
– fermesomme
Jan 25 at 11:55
$begingroup$
It is a bit strange since the author hasn't introduced homology groups, so I'm not sure what he's asking. Probably to repeat the computations in the theorem. To calculate the singular homology groups of a sphere, there are a couple of different ways. One is to make use of exact sequences of $(S^n, D^{n+})$, another uses Mayer-Vietoris and induction.
$endgroup$
– fermesomme
Jan 25 at 11:55
$begingroup$
@fermesomme Only the homology groups of a simplicial complex have been introduced.
$endgroup$
– John R. Skukalek
Jan 25 at 15:59
$begingroup$
@fermesomme Only the homology groups of a simplicial complex have been introduced.
$endgroup$
– John R. Skukalek
Jan 25 at 15:59
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%2f3084671%2fexercise-in-simplicial-homology%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%2f3084671%2fexercise-in-simplicial-homology%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
$begingroup$
What does "closure" of a simplex mean here?
$endgroup$
– fermesomme
Jan 25 at 11:41
$begingroup$
@fermesomme It means the complex that consists of all faces of the simplex.
$endgroup$
– John R. Skukalek
Jan 25 at 11:54
$begingroup$
It is a bit strange since the author hasn't introduced homology groups, so I'm not sure what he's asking. Probably to repeat the computations in the theorem. To calculate the singular homology groups of a sphere, there are a couple of different ways. One is to make use of exact sequences of $(S^n, D^{n+})$, another uses Mayer-Vietoris and induction.
$endgroup$
– fermesomme
Jan 25 at 11:55
$begingroup$
@fermesomme Only the homology groups of a simplicial complex have been introduced.
$endgroup$
– John R. Skukalek
Jan 25 at 15:59