Honors Thesis

In the age of COVID, finishing college is strange. Of course, this qualm does not compare to the livelihood- and life-threatening issues that people are dealing with, and I am not trying to complain. I was honored to graduate from UC Berkeley, thrilled to attend at all, and very happy to celebrate at home. But the completion of my honors thesis feels so anticlimactic: somehow, I would like to share this project to affirm my own work and the contributions of my amazing advisor, David Nadler. Since this blog is my main portal to share mathematical things, I am linking my thesis here. Large swaths of it just recount the brilliant exposition of Milnor's "Lectures on the H-Cobordism Theorem," which is a much better reference on the subject. However, I will make note of my favorite parts of each chapter below.

Chapter 2: The main goal of this section, as I wrote it, was to frame Morse theory as the language in which handles, surgery and (unoriented) cobordism become equivalent(ish). While much of this is present in Milnor's book (the rest is implicit) and all of it was known at the time of writing, I liked this narrative and really enjoyed the experience of deepening my own understanding by trying to write a good, intuitive explanation of these results.

Chapter 3: The third section deviates from Milnor in its discussion of Morse homology, which were not fully explicit at the time. I had read about Morse homology, but I had the misunderstanding that some infinite-dimensional methods were necessary (Sard-Smale) and that the invariance of the homology required further details than I would be able to write about. It turns out that I was wrong and that, with all the proper definitions in place, the invariance of Morse homology simply falls out of Milnor's exposition. This was both gratifying to realize and extremely enjoyable to write about. 

Chapter 4: My favorite parts of this section were the pictures (which were quite challenging to make) and one small part of one proof. In part (a) of Prop. 4.1.3, I replaced Milnor's proof using sequences with an argument based on upper semi-continuity.  It's not better per se, but I had a lot of fun coming up with it and writing it, which is worth something in itself.

Chapter 5: In the last chapter of Milnor, he discusses the h-cobordism theorem, as well as the classification of disks and spheres. With his overview and knowledge of a few seminal results that occurred since, it is not hard to fill in most of the gaps. But I didn't see where the classification of the 5-disk fit into the picture. As such, I was thrilled when I realized that it was also equivalent to the smooth Poincaré conjecture in dimension 4 (I'm sure this is well-known, but I was having trouble finding something on it). I found this interesting, because it is common to have related results in adjacent dimensions, but the statements of the two results are usually different. However, the 4D Poincaré conjecture is also equivalent to the classification of the 4-disk, so the two cases of the classification of disks that remain open ($n=4$ and $n=5$) are actually equivalent. I also learned (from a comment by Ian Agol, recounting an observation by Mike Freedman) that the following assertions are equivalent:

• The smooth 4D Schoenflies conjecture is false, i.e. there is an embedding $S^3\hookrightarrow S^4$ (where $S^4$ has the standard smooth structure) that cuts $S^4$ into two exotic $4$-disks;
• The monoid of smooth structures on $S^4$ contains some invertible element besides the standard $S^4$, i.e. there are two exotic $4$-spheres whose connected sum is standard.

The proof of this equivalence hinges on two facts: cutting a standard 4-disk out of an exotic 4-sphere yields an exotic 4-disk (this follows from Cerf's theorem on twisted 4-spheres); and gluing a standard 4-disk to an exotic 4-disk yields an exotic 4-sphere (this follows from the Cerf-Palais disk theorem).

There are certainly mistakes scattered throughout this thesis. Possibly, some of these mistakes occur in the parts I have highlighted above. While I hope not, I also hope that anyone who is inclined to read this and notices such a mistake will tell me of it. Regardless, this blog also seems like a good place to post errata:

• Near the end of p.49, I write $T\cong D^{\,n-k-1}\times S^1_{\mathscr D}(q)$. It should be $T\cong D^{\,n-k}\times S^1_{\mathscr D}(q)$, so that $\dim T=n-1=\dim V_0$. However, the bounds that are subsequently invoked have wiggle room, so the conclusions that follow remain valid.