Some thoughts on mathematics

Back for more!  Recall that we are interested in proving the following proposition:  Proposition:  There exists an irreducible, i.e. prime, homology 3-sphere which bounds a contractible 4-manifold and no bounding contractible four-manifold can be built with fewer than five handles.   In the last post, we showed that it suffices to show that there is a prime homology 3-sphere that bounds a contractible 4-manifold that is *not* surgery on a knot in $S^2 \times S^1$.  That was because being surgery on a knot in $S^2 \times S^1$ is equivalent to bounding a Mazur manifold, and those are the only contractible 4-manifolds with a handle decomposition with fewer than five handles.  That's our goal.  If you don't care about a proof of the proposition, here's a surgery picture for an example (and if you do care, use this picture as reference for the constructive proof below):   We'll start the proof by giving an obstruction to a homolo...

This sequence of posts was inspired by some conversations with Mike Miller Eismeier In this post, I want to talk about some contractible four-manifolds and connections with Dehn surgery .   Recall that lots of homology 3-spheres bound contractible four-manifolds.  For example, $S^3_{1/n}(K)$ bounds a contractible four-manifold if $K$ is a smoothly slice knot, as do the Brieskorn spheres $\Sigma(3,4,5)$ and $\Sigma(2,3,13)$.  By a quick Euler characteristic computation, it is easy to see that every contractible four-manifold other than the four-ball requires at least three handles.  In this sequence of posts I want to give a proof of the following goofy result (and give it some context).  Proposition:  There exists an irreducible, i.e. prime, homology 3-sphere which bounds a contractible 4-manifold and no bounding contractible four-manifold can be built with fewer than five handles.  Alternatively, there are prime homology 3-spheres that bound con...

It's winter break, so here's a short post.  I am trying to spend some time doing new non-math things.  The other day I was trying to help out with a fence repair.  If you have spent time with me, you'd likely think that I would not know how to help someone repair a fence - if you thought that, you are indeed correct.  I was not particularly helpful, but while I was embarrassing myself not knowing how to use a hammer, I did gain some insight into the mathematical process, which I wanted to mention here. In low-dimensional topology, as I hope to convey in this blog, it is very beneficial to have a wide range of tricks and tools.  This includes knowing topological tools (e.g. the Montesinos trick or Kirby calculus), invariants (e.g. gauge theory, Khovanov homology), but also perspectives and connections from other fields (e.g. viewing torus knots as links of singularities of algebraic curves).  However, knowing that hammers exist is not the same as being able ...

In Excision part 1, we studied how Floer homology changes when we cut and reglue along a surface.  In this post, I wanted to discuss an analogue for annular Khovanov homology.  (As a warning, I am not an expert in Khovanov homology, and definitely not knowledgeable about annular Khovanov homology.  Make sure to really fact check this!)   First, recall that given an oriented link $L$ in $S^3$, Khovanov homology $Kh(L)$ is a bigraded vector space which categorifies the Jones polynomial.  Annular Khovanov homology, $AKh$, is a *triply* graded invariant of oriented links in $D^2 \times S^1$, developed by Asaeda-Przytycki-Sikora.  The third grading is called the annular  grading, and we write $AKh(L,f)$ to mean the summand in annular grading $f$.  It's an important point that this is an invariant of the link up to isotopy in $D^2 \times S^1$ and not up to diffeomorphism.  In particular, you can cut along a meridional disk and reglue by ...

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...