Jack Brands's 2018 honours thesis in mathematics, on "Understanding the Lee Spectral Sequence", supervised by
Scott Morrison, is available here.
Outline
Khovanov homology is a (co)homology theory that gives invariants of tangles. It was originally described by Khovanov as a categorification of the Jones polynomial in [Kho00]. This original construction, however, was only defined for links, but in a series of papers [BN02] [BN05] [BN07] Bar-Natan generalised the theory to give an account for tangles, as did Khovanov in [Kho01]. At about the same time, Lee [Lee05] defined another homology theory, appropriately dubbed 'Lee homology' that is 'interestingly boring'. In fact, as we will explain below, Khovanov homology is naturally viewed as the second page of a spectral sequence that converges to Lee homology. This spectral sequence was then skilfully used by Rasmussen in [Ras10] to define the 's-invariant', which is a knot invariant that provides an obstruction to 4-dimensional smooth structure. More sepcifically, the s-invariant $s(K)$ of a knot $K$ gives a lower bound on the slice (4-ball) genus of such a knot,
$$|s(K)| \leq 2g_4(K).$$
That is, any surface $\Sigma$ in $B_4$, with $\partial \Sigma = K \subset S^3 = \partial B^4$ has minimal genus at least $\frac{1}{2}|s(K)|$. This is great news since we have very few invariants in 4 dimensional topology, and of those the interesting ones come from gauge theory which are hard to compute. For example, one consequence of the s-invariant is a purely combinatorial proof of the Milnor Conjecture [Ras10] that was originally proved using gauge theory by Kronheimer and Mrowka [KM93].
In [FGMW10], Freedman, Gompf, Morrison, and Walker showed that for a certain 4-manifold $W^4$, where $W^4$ is homeomorphic to $B^4$, but where it is unknown whether $W^4$ and $B^4$ are diffeomorphic, there is a knot $K$ in $S^3$ which is slice in $W$ so it bounds a disk in $W$. In other words, if the slice genus $g_4(K) > 0$, then $W^4$ is not diffeomorphic to $B^4$. Thus, motivating this thesis is that we need to find more 4-manifold invariants out of Khovanov homology, not just the s-invariant. We would like to have, for example, another invariant $s^W$ for knots $K \subset \partial W \neq S^3$, so that $|s^W(K)| \leq \textrm{slice-genus}_W(K)$. To do this, we will investigate the Lee spectral sequence, but this method seems much harder to generalise from $\partial B^4$ to $\partial W^4$.
