# Another Sunday, or Awaiting Week 4

3 weeks.

I’ve officially survived the first three weeks of my second year of grad school (twice, actually). Again, I know keeping count of the days is a terrible thing to do to myself, particularly when there’s been a very small amount of good to come from my weeks thus far, but at this point, I’m sort of using that countdown as some sort of badge of accomplishment. Or something.

The coming weeks are going to be very very stressful and busy and stressful. Besides my usual load of stuff (I’m enrolled in 6 classes, I have a reading class in algebraic geometry starting up on Tuesday, I’m TAing for 1 lecture and 7 labs, and I’m trying to pick advisors / plan presentations I’ll need to give some timem soon), I also thought it was a good idea (remind me why?!) to make a poster to present at FSU’s upcoming Math Fun Day. That particular endeavor shouldn’t be especially difficult, but it requires time and time, ladies and gentlemen, is precisely what I have zero of.

Daunting is the adjective that comes to mind.

Also daunting is / was / has been the thought of continuing my goal to do all the problems in Hatcher. As you may recall, I spent the first half of summer slaving to acquire the information needed for the Chapter 0 exercises, only to have my plan for Chapter 1 totality derailed by that little piece of awesome that was my Wolfram internship. Long story short: The obsessive-compulsive part of me wants to not move forward until I hash out a Chapter 1 plan, but the This will benefit me in the class I’m taking now which, subsequently, hinges on my ability to understand Chapters 2 and 3 of Hatcher part wants to press forward.

I’m pleased to announce that the second guy won out.

In particular, my Hatcher Solutions page is showing signs of progress. It didn’t take as long as I’d predicted it to take to build that framework, and due to a random, unforeseen bout of sleeplessness at 3am this morning, I had precisely the opportunity needed to seize the moment. Right now, all those are empty pages, but I’m pleased to report that I seem to have accumulated approximately six solutions; if everything goes as planned, I’ll be taking time to update by including those as soon as possible.

In the meantime, I’m going to continue to hash out what to do about this paper. And what to do about the professors I’m emailing regarding potential advisor-hood. And what to do about the fact that I severely cut my weekend work time by spending yesterday ballin’ out of control in celebration of my wife’s birth. And what to do about….

Au revoir, internet. I bid thee well.

Oh, I just remembered: I have my first exam of the semester Friday. It’s on field theory. I’m less than pleased.

# Study Plan, tentatively, + Algebraic Geometry Exercises

So I think it’s probably best to have a rotating study plan schedule that allows me to do certain topics on certain days. So far, I’m thinking of having a rotation that looks something like:

Differential Geometry -> Algebra -> Clifford Stuff -> Algebraic Topology (optional),

and since yesterday was (unofficially) differential geometry day, I’m going to spend today doing algebra.

First order of business: Eisenbud and Harris. And, since I’ve been meaning to write down some of the solutions to exercises I’ve passed, I guess I’ll do that here.

# Some Exercises from Lee’s Introduction to Smooth Manifolds

These are exercises from the first few pages of Lee’s Introduction to Smooth Manifolds. These are pretty elementary from a topology perspective – maybe the middle-to-late part of a semester on point-set topology – but I decided to put them here to (a) remember some of that stuff, and (b) force myself to not delay doing them any longer. As it turns out, I struggled more with these than I should have and even had to consult the internet for a couple thing; in instances of internet borrowing, I provide links.

Note, finally, that if you’re trying to hunt these particular exercises down within Dr. Lee’s book, they’re actually dispersed throughout the text; suffice it to say, I’m not a fan of that particular method of delivery. Nevertheless….

# Study Plan + Hatcher Section 0 solutions

So going into the summer, I had a pretty concise game plan of how I wanted things to work. I wanted (a) to exempt all my qualifying exams so that I’d have fewer have to’s on my agenda, and (b) to spend all the time I’d save through completing (a) split between (i) studies to help me with my required coursework next semester and (ii) studies to help me towards being able to research things I want to research.

Worst case scenario, of course, was that I didn’t exempt qualifying exams and had to incorporate preparation for those into the mix. As it stood, though, I had a clear idea of what my summer was going to consist of, academia-wise. I planned on taking the first week (i.e., last week) to casually do some independent research without a rigid schedule in place, which would turn into a regimented plan – beginning today and lasting until the beginning of fall – consisting of one hour of studying per day in each of the following fields:

• Clifford Analysis.
• Differential Topology/Geometry.
• Algebraic Topology.
• Abstract Algebra.

Among the four fields listed above, the first two were for me and me alone, whereas the last two were to help me for next semester.

It’s now 4:30am Tuesday. I didn’t spend Monday adhering to that regimen and, for all intents and purposes, I probably won’t today. There’s a long list of excuses I could give about how I had to go to the ER, about how my infant son is teething and angry, etc. etc., but the story is simply this: I made a regimen that I don’t seem to be able to stick to. Bing, bang, boom.

I’m going to do my best to get something constructive out of this summer, but I’m facing the sad reality that it probably won’t be what I’d planned. Boo hoo.

To aid my attempts, I’m going to try to put some of my solutions here as a way of forcing myself to stay dedicated. Somewhat. For now, here are some solutions for the Chapter 0 material from Hatcher.

Hatcher, Algebraic Topology, Chapter 0
1. Construct an explicit deformation retraction of the torus with one point deleted
onto a graph consisting of two circles intersecting in a point, namely, longitude and
meridian circles of the torus.

Proof. Let $T$ be the torus and let $X=T\setminus\{\text{pt}\}$ be the space in question. By considering the square gluing diagram of the torus $T$ sans a point, a diagrammatic representation for $X$ can be given as shown in Figure 1 below.

Figure 1

Note that the gray part of Figure 1 represents the fact that the torus $T$ is filled in and that one can visualize the deformation retract in question by grabbing hold of the hole in $X$ and stretching it out so that the diagram in Figure 1 is hollow (i.e., not filled in, i.e. all white, etc.). Thus, the deformation retract in question amounts to the projection of the interior of a square (the square representing the gluing diagram of $T$) onto its boundary, the formula for which can be attained by presupposing that the gluing diagram for $T$ is placed in $\mathbb{R}^2$ as $J\times J$ where $J=[-1,1]$. Basic arithmetic shows that, for $t\in I$, the family $f_t:I\times I\to\mathbb{R}^2$ given by

$f_t(x,y)=(1-t)(x,y)+t\left(\dfrac{(x,y)}{\max\{|x|,|y|\}}\right)$

“does the trick.” Indeed, note that $f_0(x,y)=(x,y)$, that $f_1(x,y)=(x,y)/\max\{|x|,|y|\}$ is an element of $\partial(J\times J)$, and that $f_t(x,y)|_{\partial(J\times J)}=(x,y)$ due to the fact that $\max\{|x|,|y|\}=1$ on $\partial(J\times J)$. Finally, note that continuity of the family $f_t$ is given due to the fact that $f_t$ is the composition of continuous functions of $x,y,t$ for all $t\in I$. $\,\,\square$

2. Construct an explicit deformation retraction of $\mathbb{R}^n\setminus\{0\}$ onto $S^{n-1}$.
Proof. This problem is essentially a problem from Calculus III. Note that for a vector $\mathbf{x}$ in $\mathbb{R}^n\setminus\{0\}$, the normalized vector $\mathbf{x}/\|\mathbf{x}\|$ lies on $S^{n-1}$. It suffices, then, to do the normalization process in a way that’s continuous for a time parameter $t\in I$, and one way to accomplish this is to define a family $f_t:\mathbb{R}^n\setminus\{0\}\to S^{n-1}$ so that, for each $\mathbf{x}=(x_1,x_2,\ldots,x_n)$ in the domain,

$f_t(x_1,x_2,\ldots,x_n)=\left(\dfrac{x_1}{t\|\mathbf{x}\|+(1-t)},\cdots,\dfrac{x_n}{t\|\mathbf{x}\|+(1-t)}\right).$

As noted in problem 1, the function $f_t$ is continuous for each $t\in I$. Moreover, $f_0(\mathbf{x})=\mathbf{x}$, $f_1(\mathbf{x})=\mathbf{x}/\|\mathbf{x}\|$, and $f_t(\mathbf{x})|_{S^{n-1}}=\mathbf{x}$ due to the fact that $\|\mathbf{x}\|=1$ for all $\mathbf{x}\in S^{n-1}$. $\,\,\square$