There's plenty good to do in the city if you know where to look. Our school offers student season passes to the museums and theatres. Free admission to the playhouse, though its last choice in seating.
@RobertCardona Remember: two elements of $\pi_1(X)$ are freely homotopic if and only if they're conjugate. As a corollary, if $\pi_1(X)$ is abelian, two elements are freely homotopic iff they're basepoint-homotopic.
@MikeMiller, so I was saying $f, g : I \to S^1$ are freely homotopic (which I tend to call just homotopic) but they are not basepoint-homotopic (which I call path-homotopic)?
OK, we're using the terms differently. Let me be careful here. Two maps $f,g: X \to Y$ are (freely) homotopic if there's a map $X \times I \to Y$ that restricts to $f$ and $g$ on $X \times 0$ and $X \times 1$.
Now let $(X,x), (Y,y)$ be pointed spaces. Two pointed maps, that is maps with $f(x)=y$ and $g(x)=y$, are basepoint-homotopic if there's a map $f_t: X \times I \to Y$ such that $f_0 = f, f_1 = g$ and $f_t(x)=y$.
You can generalize this to work with $I$, but the key here is that I'm choosing a basepoint on the circle (and on the space)
The only interesting thing to do with $I$ is to fix both endpoints and homotope that way. (Otherwise, because $I$ is contractible, you can just suck everything down to its basepoint.)
You're correct that any two maps $I \to S^1$ are freely homotopic. If you demand that the homotopies fix eg $f(0)$, then every map $I \to S^1$ with $g(0)=f(0)$ is homotopic.
If you demand that the homotopies also fix $f(0)$ and $f(1)$ you get plenty that are not homotopic.
That was my original question. I never came across this, surprisingly, when I took a class in algebraic topology, but it came up in a problem I was looking at this morning.
That's what I was talking about earlier. One calculates the fundamental group $\pi_1(S^1,1) = \Bbb Z$. So there are plenty of different non-basepoint-homotopic maps to $S^1$.
It's a general fact - not hard to prove - that elements of $\pi_1(X,x_0) are freely homotopic iff they're conjugate in $\pi_1$.
Because $\pi_1(S^1)$ is abelian, elements are conjugate if and only if they're equal; so two (basepointed) maps are homotopic iff they're basepoint homotopic.
Well, I say that since I will be taking algebraic topology in the spring, so I'm trying to watch through youtube lectures on the fundamental group to try to get a head start
Well, I can't tell you the table of contents, since I don't have 'em. Topics I remember from the course: the fundamental group, Toeplitz operators, Gauss-Bonnet, and Bott periodicity.
yes, @pourjour, and the number of nonzero rows and number of nonzero columns agree
I have had plenty of undergraduates who excelled at my graduate differential geometry course, @Mike. That didn't make the course an undergraduate course or a course accessible to most undergraduates.
But I'll be curious to see his book (if one can still find books somewhere without buying them)
@LeGrandDODOM sorry to hear that, cause i learned my french using the book "french for dummies" and my english using the book "L'Anglais pour les nuls"
@LeGrandDODOM to say the words of truth, i have a french friend on facebook, i told him to write some sentences in french so that i can make as if i'm communicating with you but he was unfortunately smarter
@evinda Hoho! thhanks! happy new year!! best wishes for ya : )
@AlecTeal it's because of this: "How about you wait with posting the question until you can post something that makes sense? " anyway i gave you an upvote : )
@user153330 Ah, do you know what this year reminds me of? (you can guess because I've been talking about this year for more than 2 years and I've been thinking about it for more than 13, some people have been thinking about it for 30 years!!)
@JMoravitz I just made that function up right now. I can write that function using a for loop in a programming language such as Javascript, C++, python, etc but I don't know how to mathematically define it. Any ideas?
you could define it using a sum of floors of negative powers of 10 times the number ranging from 0 to the log base 10 of the number, but it seems rather contrived
@MikeMiller I never called Daniel a politician, as far as you know I was sharing a moral question that could have been totally disattached from any reference to Daniel.
@MikeMiller Don't stick my previous statement in any equation. It's a hyperbole where the monarch butterfly's lifespan (2-6 weeks) is a reference to a small quantity. Otherwise, that statement is dimensionally inconsistant as I'd be equating neural capacity with time.
@BalarkaSen A brief review of 2014: Robin Williams died. Grothendieck died. Two Malaysian Airlines planes went missing and a ton of irrelevance which no one cares for anymore.
@MikeMiller No, that's pretty much how he dies. The story folks fixed his death. The normal teenage adventures will continue but the future is fixed. Just like our lives.
@JasperLoy My words are open to interpretation. You don't want to get into a debate with me about my beliefs. I don't have any other than my solid morality and my deep philosophies which I mutate based on the audience.
@BalarkaSen Seriously, you do know how he dies, right? That's pretty heroic. A sort of hero cliche but I always admire that kind of "put the needs of another before yourself" kind of moments.
@MikeMiller To be honest, I don't remember it either but it sounds like a more standard answer than Pandora and I dare not use Uranus again because of the jokes it may spawn.
@BalarkaSen It's not really british and also, whenever I address you as something, I intend for a level of respect to conveyed. I didn't do it because the Queen told me to. kay?
@JasperLoy I have never had a conversation in this room that didn't in some way or another reference a fruit. Most oftenly, it is a direct reference through the word "banana". Why is everything bananas here?
