Showing that $mathbb{C}$ minus a point is homotopy equivalent to $S^1$$X$ and $Y$ are homotopy equivalent...
Should I take out a loan for a friend to invest on my behalf?
Why does Deadpool say "You're welcome, Canada," after shooting Ryan Reynolds in the end credits?
Should QA ask requirements to developers?
Are babies of evil humanoid species inherently evil?
Is having access to past exams cheating and, if yes, could it be proven just by a good grade?
Is "history" a male-biased word ("his+story")?
Do f-stop and exposure time perfectly cancel?
How did Alan Turing break the enigma code using the hint given by the lady in the bar?
Examples of a statistic that is not independent of sample's distribution?
Things to avoid when using voltage regulators?
Why does the negative sign arise in this thermodynamic relation?
They call me Inspector Morse
Could you please stop shuffling the deck and play already?
Am I not good enough for you?
Low budget alien movie about the Earth being cooked
BitNot does not flip bits in the way I expected
Why don't MCU characters ever seem to have language issues?
Can you reject a postdoc offer after the PI has paid a large sum for flights/accommodation for your visit?
How to pass a string to a command that expects a file?
What Happens when Passenger Refuses to Fly Boeing 737 Max?
A question on the ultrafilter number
Why would one plane in this picture not have gear down yet?
MTG: Can I kill an opponent in response to lethal activated abilities, and not take the damage?
What are some noteworthy "mic-drop" moments in math?
Showing that $mathbb{C}$ minus a point is homotopy equivalent to $S^1$
$X$ and $Y$ are homotopy equivalent $Leftrightarrow$ $exists Z:$ $X,Y$ are strong deformation retracts of $Z$Homotopy equivalence in the category of arrows.Showing that a homotopy fiber of a fibration is homotopy equivalent to the fiber of the base point.Exercise 2, chapter 4, Hatcher.Showing that $mathbb S^1$ is a deformation retract of the Mobius strip, rigorously.$X$ is contractible if and only if $X simeq { * } $ - A three-part questionWhich surface is homotopy equivalent to $Bbb{R}^4$ minus the planes $x=y=0$, $z=w=0$?Prove the cylinder $S^1 times I$ is homotopy equivalent to the circle $S^1$Is $(I^n, partial I^n)$ homotopy equivalent to $(R^n, R^nsetminus left{0right})$Does a deck transformation have a homotopy that lifts to it?
$begingroup$
Intuitively this is clear (I think)? However I struggled to construct the homotopy. Here's what I attempted:
Define $M:=mathbb{C} - p$ where $p$ is a point in $mathbb{C}$.
Define also, $alpha:S^1rightarrow M$, given by $x mapsto x$.
And Define $beta:Mrightarrow S^1$, given by $ymapsto frac{y}{mid y mid}$.
This gives us $betaalpha(x) = Id_{S^1}$. So all that is left to show is that and $alphabeta(x) = frac{x}{mid x mid} simeq Id_{M}$. Here's where I struggled to come up with a homotopy and perhaps that is because I picked $alpha$ and $beta$ wrong to start with?
general-topology homotopy-theory
asked Mar 9 at 16:52
PrimebrookPrimebrook
605
$endgroup$
add a comment |
$begingroup$
Intuitively this is clear (I think)? However I struggled to construct the homotopy. Here's what I attempted:
Define $M:=mathbb{C} - p$ where $p$ is a point in $mathbb{C}$.
Define also, $alpha:S^1rightarrow M$, given by $x mapsto x$.
And Define $beta:Mrightarrow S^1$, given by $ymapsto frac{y}{mid y mid}$.
This gives us $betaalpha(x) = Id_{S^1}$. So all that is left to show is that and $alphabeta(x) = frac{x}{mid x mid} simeq Id_{M}$. Here's where I struggled to come up with a homotopy and perhaps that is because I picked $alpha$ and $beta$ wrong to start with?
general-topology homotopy-theory
asked Mar 9 at 16:52
PrimebrookPrimebrook
605
$endgroup$
1
$begingroup$
Note that $|z|in(0,infty)$ for $zin Bbb C^times$. Can you show that $(0,infty)$ is homotopy equivalent to ${1}$?
$endgroup$
– Hagen von Eitzen
Mar 9 at 16:57
add a comment |
$begingroup$
Intuitively this is clear (I think)? However I struggled to construct the homotopy. Here's what I attempted:
Define $M:=mathbb{C} - p$ where $p$ is a point in $mathbb{C}$.
Define also, $alpha:S^1rightarrow M$, given by $x mapsto x$.
And Define $beta:Mrightarrow S^1$, given by $ymapsto frac{y}{mid y mid}$.
This gives us $betaalpha(x) = Id_{S^1}$. So all that is left to show is that and $alphabeta(x) = frac{x}{mid x mid} simeq Id_{M}$. Here's where I struggled to come up with a homotopy and perhaps that is because I picked $alpha$ and $beta$ wrong to start with?
general-topology homotopy-theory
asked Mar 9 at 16:52
PrimebrookPrimebrook
605
$endgroup$
Intuitively this is clear (I think)? However I struggled to construct the homotopy. Here's what I attempted:
Define $M:=mathbb{C} - p$ where $p$ is a point in $mathbb{C}$.
Define also, $alpha:S^1rightarrow M$, given by $x mapsto x$.
And Define $beta:Mrightarrow S^1$, given by $ymapsto frac{y}{mid y mid}$.
This gives us $betaalpha(x) = Id_{S^1}$. So all that is left to show is that and $alphabeta(x) = frac{x}{mid x mid} simeq Id_{M}$. Here's where I struggled to come up with a homotopy and perhaps that is because I picked $alpha$ and $beta$ wrong to start with?
general-topology homotopy-theory
general-topology homotopy-theory
asked Mar 9 at 16:52
PrimebrookPrimebrook
605
asked Mar 9 at 16:52
PrimebrookPrimebrook
605
edited Mar 9 at 17:01
Primebrook
asked Mar 9 at 16:52
PrimebrookPrimebrook
605
asked Mar 9 at 16:52
PrimebrookPrimebrook
605
asked Mar 9 at 16:52
PrimebrookPrimebrook
605
605
1
$begingroup$
Note that $|z|in(0,infty)$ for $zin Bbb C^times$. Can you show that $(0,infty)$ is homotopy equivalent to ${1}$?
$endgroup$
– Hagen von Eitzen
Mar 9 at 16:57
add a comment |
1
$begingroup$
Note that $|z|in(0,infty)$ for $zin Bbb C^times$. Can you show that $(0,infty)$ is homotopy equivalent to ${1}$?
$endgroup$
– Hagen von Eitzen
Mar 9 at 16:57
1
1
$begingroup$
Note that $|z|in(0,infty)$ for $zin Bbb C^times$. Can you show that $(0,infty)$ is homotopy equivalent to ${1}$?
$endgroup$
– Hagen von Eitzen
Mar 9 at 16:57
$begingroup$
Note that $|z|in(0,infty)$ for $zin Bbb C^times$. Can you show that $(0,infty)$ is homotopy equivalent to ${1}$?
$endgroup$
– Hagen von Eitzen
Mar 9 at 16:57
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: Given two points $p,q in mathbb C$, how do you parameterize the line segment $overline{pq}$ using the parameter interval $0 le t le 1$?
answered Mar 9 at 17:02
Lee MosherLee Mosher
50.5k33787
$endgroup$
$begingroup$
Thank you, perfect answer. It's made it obvious now. :)
$endgroup$
– Primebrook
Mar 9 at 17:08
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%2f3141331%2fshowing-that-mathbbc-minus-a-point-is-homotopy-equivalent-to-s1%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
$begingroup$
Hint: Given two points $p,q in mathbb C$, how do you parameterize the line segment $overline{pq}$ using the parameter interval $0 le t le 1$?
answered Mar 9 at 17:02
Lee MosherLee Mosher
50.5k33787
$endgroup$
$begingroup$
Thank you, perfect answer. It's made it obvious now. :)
$endgroup$
– Primebrook
Mar 9 at 17:08
add a comment |
$begingroup$
Hint: Given two points $p,q in mathbb C$, how do you parameterize the line segment $overline{pq}$ using the parameter interval $0 le t le 1$?
answered Mar 9 at 17:02
Lee MosherLee Mosher
50.5k33787
$endgroup$
$begingroup$
Thank you, perfect answer. It's made it obvious now. :)
$endgroup$
– Primebrook
Mar 9 at 17:08
add a comment |
$begingroup$
Hint: Given two points $p,q in mathbb C$, how do you parameterize the line segment $overline{pq}$ using the parameter interval $0 le t le 1$?
answered Mar 9 at 17:02
Lee MosherLee Mosher
50.5k33787
$endgroup$
Hint: Given two points $p,q in mathbb C$, how do you parameterize the line segment $overline{pq}$ using the parameter interval $0 le t le 1$?
answered Mar 9 at 17:02
Lee MosherLee Mosher
50.5k33787
answered Mar 9 at 17:02
Lee MosherLee Mosher
50.5k33787
answered Mar 9 at 17:02
Lee MosherLee Mosher
50.5k33787
answered Mar 9 at 17:02
Lee MosherLee Mosher
50.5k33787
50.5k33787
$begingroup$
Thank you, perfect answer. It's made it obvious now. :)
$endgroup$
– Primebrook
Mar 9 at 17:08
add a comment |
$begingroup$
Thank you, perfect answer. It's made it obvious now. :)
$endgroup$
– Primebrook
Mar 9 at 17:08
$begingroup$
Thank you, perfect answer. It's made it obvious now. :)
$endgroup$
– Primebrook
Mar 9 at 17:08
$begingroup$
Thank you, perfect answer. It's made it obvious now. :)
$endgroup$
– Primebrook
Mar 9 at 17:08
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%2f3141331%2fshowing-that-mathbbc-minus-a-point-is-homotopy-equivalent-to-s1%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
1
$begingroup$
Note that $|z|in(0,infty)$ for $zin Bbb C^times$. Can you show that $(0,infty)$ is homotopy equivalent to ${1}$?
$endgroup$
– Hagen von Eitzen
Mar 9 at 16:57