Some thoughts on mathematics

In algebraic topology, the excision theorem is an extremely useful tool for computing homology.  In case your algebraic topology class is ancient history, let's recall the statement: if $U \subset V \subset X$ are a sequence of spaces with $\overline{U} \subset V^\circ$, then $$H_*(X - U, A - U) \cong H_*(X, A).$$In other words, relative homology is unaffected by "cutting" out $U$.  In Floer homology, we also have excision, which involves a different kind of cutting.  In this post I want to discuss some aspects of Floer homology in excision.  In the next post, I'll give proof of a similar statement in annular Khovanov homology.     Simplest version of excision:  First I want to state and prove an easy version of Floer excision.  I'm going to write this in a way that applies in most Floer theories, so I'll use $F$ to denote an arbitrary Floer homology.  Let $Y$ be a closed connected oriented three-manifold.  Suppose that $x$ is...

There is an upcoming conference celebrating Cameron Gordon's 80th birthday.  Cameron has had a big influence on my career (both directly and through his mathematics), so I wanted to write some posts giving quick summaries of his influential work.  These will be short samples instead of giving full details on anything.  This post was inspired by a comment of Josh Wang. I thought it might be kind of cliche to start with the most famous results, like the Knot Complement Theorem, so I'll instead start with one of my personal favorites: "Only integral surgeries can yield reducible manifolds" by Gordon-Luecke.  While the title is the result, we'll see some more background and at the end, I'll give a really nice application to the unknotting number of knots.   First, we need to have a little chat about Dehn surgeries.  (As an aside, Cameron is also a historian of topology.  Since Dehn came up, I'll point out Cameron wrote a historical arti...

I don't necessarily think the various areas I discuss below are more or less important than the others, just what I view as their popularity.  Others likely have a different view.  Recently, I was listening to The Beths' "Expert in a Dying Field".  It's a break-up song about all the minutia you learn about a person which no longer feels relevant when the relationship ends.  (Don't worry, I'm not going through a breakup/angst phase.  I just like the song.)  When I listen to it, I often times think about Heegaard Floer homology.   Heegaard Floer homology, developed by Ozsvath and Szabo, exploded in the early 2000s.  It seemed to pack most of the punch of gauge theory while being super computable.  When I started grad school there was this incredible flurry of activity, both in theory building (e.g. bordered Floer homology) and applications (e.g. structure of the knot concordance group, classifying fillable contact structures).  Hot fi...

Here's a short post on a construction I really enjoy.   Theorem: (Kawauchi)  Let $K \subset S^3$ be a strongly negative amphichiral knot (SNACK).  Then $K$ bounds a smoothly embedded disk in a rational homology ball.   Let me first explain the statement and then give a proof.  A knot $K$ is strongly negative amphichiral if there is an orientation-reversing diffeomorphism of $(S^3,K)$ which is also orientation-reversing on $K$ and the fixed point set is two points which sit on $K$.  Sometimes $K$ can be slice, e.g. the square knot is a slice SNACK, but often it is not, e.g. the figure-eight knot.  (The Fox-Milnor condition shows it is not even topologically slice.)   Here is the proof, which is very quick.  First, $X_0(K)$ denote a 0-framed 2-handle attachment to $B^4$ along $K$.  Then $K$ is smoothly slice in this four-manifold - the slice disk is just the core of the 2-handle!  (This applies even if $K$ is not SNACK...

Since exotic four-manifolds are always "in", I wanted to talk about a cool construction of exotica I like due to Akbulut-Ruberman.  In their paper, they construct compact contractible four-manifolds which are "absolutely exotic", which means they are homeomorphic, but there is no diffeomorphism between them.  To clarify the terminology, often times, for manifolds with boundary, one talks about "relatively exotic" manifolds, which means that you have a homeomorphism from one four-manifold to another, but the boundary homeomorphism doesn't extend to a diffeomorphism of the four-manifolds.  [For context, it's worth pointing out that long before, we know how to construct lots of non-compact, contractible exotica: namely exotic $\mathbb{R}^4$'s.  In fact, Taubes proved there are uncountably many exotic $\mathbb{R}^4$'s.] I want to describe their work (but with a bit of a Floer homology lean).   For simplicity, let's construct a pair of n...