Geometric presentation of fundamental group of a surface
$begingroup$
Let $S = S_g$ be a closed surface.
An author of a paper writes:
We say $langle a_1, b_1, cdots, a_{2g}, b_{2g} | R rangle$ is a geometric presentation of the fundamental group $pi_1(S)$ if the corresponding one vertex 2-complex is homeomorphic to S
Could somebody clarify exactly what this means?
I am familiar with the canonical presentation of $pi_1(S)$,
$$pi_1(S) = langle a_1, b_1, cdots, a_{2g}, b_{2g} | [a_1,b_1] cdots [a_{2g}, b_{2g}] = 1 rangle $$
and how it stems from idenitifying the sides of a hyperbolic 4g-gon, which labelled so that they spell out the product of the commutators $[a_1,b_1] cdots [a_{2g}, b_{2g}]$, however I'm not sure how that translates to the definition above.
algebraic-topology surfaces fundamental-groups low-dimensional-topology
asked Dec 17 '18 at 9:51
38917803891780
856
$endgroup$
add a comment |
$begingroup$
Let $S = S_g$ be a closed surface.
An author of a paper writes:
We say $langle a_1, b_1, cdots, a_{2g}, b_{2g} | R rangle$ is a geometric presentation of the fundamental group $pi_1(S)$ if the corresponding one vertex 2-complex is homeomorphic to S
Could somebody clarify exactly what this means?
I am familiar with the canonical presentation of $pi_1(S)$,
$$pi_1(S) = langle a_1, b_1, cdots, a_{2g}, b_{2g} | [a_1,b_1] cdots [a_{2g}, b_{2g}] = 1 rangle $$
and how it stems from idenitifying the sides of a hyperbolic 4g-gon, which labelled so that they spell out the product of the commutators $[a_1,b_1] cdots [a_{2g}, b_{2g}]$, however I'm not sure how that translates to the definition above.
algebraic-topology surfaces fundamental-groups low-dimensional-topology
asked Dec 17 '18 at 9:51
38917803891780
856
$endgroup$
add a comment |
$begingroup$
Let $S = S_g$ be a closed surface.
An author of a paper writes:
We say $langle a_1, b_1, cdots, a_{2g}, b_{2g} | R rangle$ is a geometric presentation of the fundamental group $pi_1(S)$ if the corresponding one vertex 2-complex is homeomorphic to S
Could somebody clarify exactly what this means?
I am familiar with the canonical presentation of $pi_1(S)$,
$$pi_1(S) = langle a_1, b_1, cdots, a_{2g}, b_{2g} | [a_1,b_1] cdots [a_{2g}, b_{2g}] = 1 rangle $$
and how it stems from idenitifying the sides of a hyperbolic 4g-gon, which labelled so that they spell out the product of the commutators $[a_1,b_1] cdots [a_{2g}, b_{2g}]$, however I'm not sure how that translates to the definition above.
algebraic-topology surfaces fundamental-groups low-dimensional-topology
asked Dec 17 '18 at 9:51
38917803891780
856
$endgroup$
Let $S = S_g$ be a closed surface.
An author of a paper writes:
We say $langle a_1, b_1, cdots, a_{2g}, b_{2g} | R rangle$ is a geometric presentation of the fundamental group $pi_1(S)$ if the corresponding one vertex 2-complex is homeomorphic to S
Could somebody clarify exactly what this means?
I am familiar with the canonical presentation of $pi_1(S)$,
$$pi_1(S) = langle a_1, b_1, cdots, a_{2g}, b_{2g} | [a_1,b_1] cdots [a_{2g}, b_{2g}] = 1 rangle $$
and how it stems from idenitifying the sides of a hyperbolic 4g-gon, which labelled so that they spell out the product of the commutators $[a_1,b_1] cdots [a_{2g}, b_{2g}]$, however I'm not sure how that translates to the definition above.
algebraic-topology surfaces fundamental-groups low-dimensional-topology
algebraic-topology surfaces fundamental-groups low-dimensional-topology
asked Dec 17 '18 at 9:51
38917803891780
856
asked Dec 17 '18 at 9:51
38917803891780
856
asked Dec 17 '18 at 9:51
38917803891780
856
asked Dec 17 '18 at 9:51
38917803891780
856
asked Dec 17 '18 at 9:51
38917803891780
856
856
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The definition of Cayley complex helps.
Briefly, given a presentation $langle x_1,...,x_n|R_1,...,R_krangle $ of a group, you construct a 2-dim complex as folows:
i) take a single vertex $v$
ii) attach to $v$ a loop $gamma_i$ for any generator $x_i$
iii) for any relation of the form $x_{i_1}cdots x_{i_s}=1$ attach a polygon with $s$ egdes the complex, by labeling edges $e_1,...,e_s$ and attaching $e_r$ to $x_{i_r}$
The result is the Cayley complex. Now, given $S$ and a presentation of $pi_1(S)$ you wonder if the Cayley complex is homeomorphic to $S$ or not.
If yes, the author of the paper you are reading call such a presentation geometric.
An example of a non-geometric presentation could be constructed for instance by artificially adding generators and relations that kills them.
answered Dec 17 '18 at 13:37
user126154user126154
5,376816
$endgroup$
add a comment |
$begingroup$
The $4g$-gon with side identifications that you mention is exactly a one-vertex $2$-complex which is homeomorphic to $S$. So set $R = [a_1,b_1]cdots [a_{2g},b_{2g}]$ and you're done.
answered Dec 17 '18 at 13:20
Dan RustDan Rust
22.9k114984
$endgroup$
$begingroup$
Hi Dan, thank you for your answer. What I really want to know is what other relations are allowed? Also, what would make a presentation of $pi_1(S)$ non-geometric?
$endgroup$
– 3891780
Dec 17 '18 at 13:28
$begingroup$
Ah sorry, your question wasn't too clear on that point. The other user's answer involving the Cayley complex is probably what you're looking for.
$endgroup$
– Dan Rust
Dec 17 '18 at 13:40
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%2f3043743%2fgeometric-presentation-of-fundamental-group-of-a-surface%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The definition of Cayley complex helps.
Briefly, given a presentation $langle x_1,...,x_n|R_1,...,R_krangle $ of a group, you construct a 2-dim complex as folows:
i) take a single vertex $v$
ii) attach to $v$ a loop $gamma_i$ for any generator $x_i$
iii) for any relation of the form $x_{i_1}cdots x_{i_s}=1$ attach a polygon with $s$ egdes the complex, by labeling edges $e_1,...,e_s$ and attaching $e_r$ to $x_{i_r}$
The result is the Cayley complex. Now, given $S$ and a presentation of $pi_1(S)$ you wonder if the Cayley complex is homeomorphic to $S$ or not.
If yes, the author of the paper you are reading call such a presentation geometric.
An example of a non-geometric presentation could be constructed for instance by artificially adding generators and relations that kills them.
answered Dec 17 '18 at 13:37
user126154user126154
5,376816
$endgroup$
add a comment |
$begingroup$
The definition of Cayley complex helps.
Briefly, given a presentation $langle x_1,...,x_n|R_1,...,R_krangle $ of a group, you construct a 2-dim complex as folows:
i) take a single vertex $v$
ii) attach to $v$ a loop $gamma_i$ for any generator $x_i$
iii) for any relation of the form $x_{i_1}cdots x_{i_s}=1$ attach a polygon with $s$ egdes the complex, by labeling edges $e_1,...,e_s$ and attaching $e_r$ to $x_{i_r}$
The result is the Cayley complex. Now, given $S$ and a presentation of $pi_1(S)$ you wonder if the Cayley complex is homeomorphic to $S$ or not.
If yes, the author of the paper you are reading call such a presentation geometric.
An example of a non-geometric presentation could be constructed for instance by artificially adding generators and relations that kills them.
answered Dec 17 '18 at 13:37
user126154user126154
5,376816
$endgroup$
add a comment |
$begingroup$
The definition of Cayley complex helps.
Briefly, given a presentation $langle x_1,...,x_n|R_1,...,R_krangle $ of a group, you construct a 2-dim complex as folows:
i) take a single vertex $v$
ii) attach to $v$ a loop $gamma_i$ for any generator $x_i$
iii) for any relation of the form $x_{i_1}cdots x_{i_s}=1$ attach a polygon with $s$ egdes the complex, by labeling edges $e_1,...,e_s$ and attaching $e_r$ to $x_{i_r}$
The result is the Cayley complex. Now, given $S$ and a presentation of $pi_1(S)$ you wonder if the Cayley complex is homeomorphic to $S$ or not.
If yes, the author of the paper you are reading call such a presentation geometric.
An example of a non-geometric presentation could be constructed for instance by artificially adding generators and relations that kills them.
answered Dec 17 '18 at 13:37
user126154user126154
5,376816
$endgroup$
The definition of Cayley complex helps.
Briefly, given a presentation $langle x_1,...,x_n|R_1,...,R_krangle $ of a group, you construct a 2-dim complex as folows:
i) take a single vertex $v$
ii) attach to $v$ a loop $gamma_i$ for any generator $x_i$
iii) for any relation of the form $x_{i_1}cdots x_{i_s}=1$ attach a polygon with $s$ egdes the complex, by labeling edges $e_1,...,e_s$ and attaching $e_r$ to $x_{i_r}$
The result is the Cayley complex. Now, given $S$ and a presentation of $pi_1(S)$ you wonder if the Cayley complex is homeomorphic to $S$ or not.
If yes, the author of the paper you are reading call such a presentation geometric.
An example of a non-geometric presentation could be constructed for instance by artificially adding generators and relations that kills them.
answered Dec 17 '18 at 13:37
user126154user126154
5,376816
edited Dec 30 '18 at 11:04
answered Dec 17 '18 at 13:37
user126154user126154
5,376816
answered Dec 17 '18 at 13:37
user126154user126154
5,376816
answered Dec 17 '18 at 13:37
user126154user126154
5,376816
5,376816
add a comment |
add a comment |
$begingroup$
The $4g$-gon with side identifications that you mention is exactly a one-vertex $2$-complex which is homeomorphic to $S$. So set $R = [a_1,b_1]cdots [a_{2g},b_{2g}]$ and you're done.
answered Dec 17 '18 at 13:20
Dan RustDan Rust
22.9k114984
$endgroup$
$begingroup$
Hi Dan, thank you for your answer. What I really want to know is what other relations are allowed? Also, what would make a presentation of $pi_1(S)$ non-geometric?
$endgroup$
– 3891780
Dec 17 '18 at 13:28
$begingroup$
Ah sorry, your question wasn't too clear on that point. The other user's answer involving the Cayley complex is probably what you're looking for.
$endgroup$
– Dan Rust
Dec 17 '18 at 13:40
add a comment |
$begingroup$
The $4g$-gon with side identifications that you mention is exactly a one-vertex $2$-complex which is homeomorphic to $S$. So set $R = [a_1,b_1]cdots [a_{2g},b_{2g}]$ and you're done.
answered Dec 17 '18 at 13:20
Dan RustDan Rust
22.9k114984
$endgroup$
$begingroup$
Hi Dan, thank you for your answer. What I really want to know is what other relations are allowed? Also, what would make a presentation of $pi_1(S)$ non-geometric?
$endgroup$
– 3891780
Dec 17 '18 at 13:28
$begingroup$
Ah sorry, your question wasn't too clear on that point. The other user's answer involving the Cayley complex is probably what you're looking for.
$endgroup$
– Dan Rust
Dec 17 '18 at 13:40
add a comment |
$begingroup$
The $4g$-gon with side identifications that you mention is exactly a one-vertex $2$-complex which is homeomorphic to $S$. So set $R = [a_1,b_1]cdots [a_{2g},b_{2g}]$ and you're done.
answered Dec 17 '18 at 13:20
Dan RustDan Rust
22.9k114984
$endgroup$
The $4g$-gon with side identifications that you mention is exactly a one-vertex $2$-complex which is homeomorphic to $S$. So set $R = [a_1,b_1]cdots [a_{2g},b_{2g}]$ and you're done.
answered Dec 17 '18 at 13:20
Dan RustDan Rust
22.9k114984
answered Dec 17 '18 at 13:20
Dan RustDan Rust
22.9k114984
answered Dec 17 '18 at 13:20
Dan RustDan Rust
22.9k114984
answered Dec 17 '18 at 13:20
Dan RustDan Rust
22.9k114984
22.9k114984
$begingroup$
Hi Dan, thank you for your answer. What I really want to know is what other relations are allowed? Also, what would make a presentation of $pi_1(S)$ non-geometric?
$endgroup$
– 3891780
Dec 17 '18 at 13:28
$begingroup$
Ah sorry, your question wasn't too clear on that point. The other user's answer involving the Cayley complex is probably what you're looking for.
$endgroup$
– Dan Rust
Dec 17 '18 at 13:40
add a comment |
$begingroup$
Hi Dan, thank you for your answer. What I really want to know is what other relations are allowed? Also, what would make a presentation of $pi_1(S)$ non-geometric?
$endgroup$
– 3891780
Dec 17 '18 at 13:28
$begingroup$
Ah sorry, your question wasn't too clear on that point. The other user's answer involving the Cayley complex is probably what you're looking for.
$endgroup$
– Dan Rust
Dec 17 '18 at 13:40
$begingroup$
Hi Dan, thank you for your answer. What I really want to know is what other relations are allowed? Also, what would make a presentation of $pi_1(S)$ non-geometric?
$endgroup$
– 3891780
Dec 17 '18 at 13:28
$begingroup$
Hi Dan, thank you for your answer. What I really want to know is what other relations are allowed? Also, what would make a presentation of $pi_1(S)$ non-geometric?
$endgroup$
– 3891780
Dec 17 '18 at 13:28
$begingroup$
Ah sorry, your question wasn't too clear on that point. The other user's answer involving the Cayley complex is probably what you're looking for.
$endgroup$
– Dan Rust
Dec 17 '18 at 13:40
$begingroup$
Ah sorry, your question wasn't too clear on that point. The other user's answer involving the Cayley complex is probably what you're looking for.
$endgroup$
– Dan Rust
Dec 17 '18 at 13:40
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%2f3043743%2fgeometric-presentation-of-fundamental-group-of-a-surface%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
