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Topology · Blog Entry
Today was pretty busy! I got up, got ready quickly, and served myself some waffles at the RoMa dining hall today. Then, at around 9 am, I walked over to Jadwin Hall and entered my PHY 105 precept. I tried to finish my Latin homework there, but class began, so I started to take notes instead.
I was pretty much on fire with answering the questions today. It was mostly a lecture but my instructor asked some questions to the class as well. We covered a review of what had been covered in the last lecture, and we looked at some examples, such as uniform circular motion, non-uniform circular motion, and "ant on a disk." I will be posting my notes for the week in the physics tab, and updating them periodically.
After that, I completed my Latin homework in the Fine Hall common room, and I compared physics problem set answers with my friend Petr. I made an admittedly embarrassing mistake where, for part c) of problem 2, I set $t=1$ like in part a) of the problem instead of $t=\pi/4$ like in part b) of the problem (it said to set $t$ accordingly based on the last part). We had a question regarding whether we were expected to write negative solutions to a certain question or not. I asked this later in office hours for physics.
Then, I went back to the RoMa dining hall for lunch. I frequent the RoMa dining hall because I live in Rockefeller college, by the way. I got myself some nice grilled chicken. After that, I decided to keep self-studying Bredon and got through the rest of the section on homotopy and started the section on topological groups. I did not cover as much as I wanted, but that is fine.
In the early afternoon hours, I walked over to the Lewis Thomas Laboratory, which is where the molecular biology department is located. I had some trouble locating my advisor's office (my advisor is Professor Yibin Kang, by the way), but I was able to with the help of some upperclassman or graduate student, I'm not sure which. He quickly checked in with me about classes, extracurriculars, how I'm liking life at Princeton, etc. To be honest, it's going great. But I am getting worried as the academics are ramping up.
At around 2 pm, I went to Fine Hall room 601 for topology office hours. There, I asked a couple of questions. The first question was about a part of a question in the problem set asking me to find a choice function for the set $X^\omega$, where $X=\{0,1\}$. Professor Gabai thought for a bit, and said something about the well-ordering of the reals. I didn't get much from that. On the other hand, his hint was immensely insightful for my other question, which was about proving that the set $X_0$ of all elements in $S_\Omega$ without an immediate predecessor is uncountable. He said to look at a specific theorem which stated that a countable subset in $S_\Omega$ has an upper bound. To be honest, the resulting proof I thought of for that question was rather involved, and required a pretty interesting argument for the countability of a certain set in order to reach a contradiction, but whatever. I'm going to be revising all my proofs this weekend.
Then, I went to office hours for physics. I was told to include both positive and negative solutions regarding my question earlier, which I did. Since I basically finished the physics problem set, I mostly just hung around and socialized with some of my friends who are also taking PHY 105, such as Brian (he went to the same high school as me), Rohan, and Lucas. After that, we went to the Princeton Society of Physics Students meeting, where we just socialized with each other.
I didn't do much after that. I ate dinner, and then I relaxed and showered. I saw a cockroach crawling about in the bathroom, which freaked me out a little bit, but whatever, I guess.
Overall, the first week went pretty well, I'd say. I am nervous that I will get a bad grade in the writing seminar. I am also nervous that the physics class will ramp up to a level which I am unable to comprehend (it is rather slow right now). But I hope topology won't be too bad, and I think Latin is going to be supremely easy. The weekend will provide a nice respite, though I will have to work on stuff for my writing seminar then. Aside from that, it'll be chill, and I hope to have plenty of time to socialize and study. I'll keep you guys updated.
Based on the Reparametrization Lemma from yesterday's blog entry, we obtain the following proposition:
We necessarily have that $F * C \simeq F \text{ rel } X \times \partial I$, and we also have $C *F \simeq F \text{ rel }X \times \partial I$
Using the Reparametrization Lemma, for the first case, let $\phi_1(t)=2t$ when $t\leq 1/2$ and 1 when $t \geq 1/2$, and $\phi_2(t)=t$, and for the second case, let $\phi_1(t)=0$ when $t\leq 1/2$, and $2t-1$ when $t\geq 1/2$, and then $\phi_2(t)=t$.
We can also define an "inverse homotopy," which essentially reverses the homotopy in question, like so:
Suppose we have a homotopy $F: X \times I \to Y$. Define $F^{-1}:X \times I \to Y$ where $F^{-1}(x,t)=F(x,1-t)$.
If $F$ is a homotopy, then $F * F^{-1} \simeq C \text{ rel }X \times \partial I$ where $C(x,t)=F(x,0)$ for all $x$ and $t$.
Let $\phi_1(t)=2t$ for $t\leq 1/2$, $\phi_1(t)=2-2t$ for $t\geq 1/2$, and $\phi_2(t)=0$ for all $t$. Then apply the Reparametrization Lemma. It follows that the two homotopies are homotopy equivalent.
The notion of a topological group is profoundly important in the fields of topology and geometry. We will now begin to look at these mathematical objects.
A topological group is a Hausdorff topological space $G$ with a group structure on $G$ such that we have the following: