Minerva Course Fall 2025

Surfaces and the Fourth Dimension

Here is a tentative plan for the Minerva Mini-Course at Princeton in Fall 2025. This page will be updated as the plan evolves. The goal of the lectures is to introduce some recent techniques in low-dimensional topology, and some famous old theorems and less famous new ones they can be used to prove.

The first talk will be at a gentle colloquium level; most of it should be accessible to second-year undergraduate mathematics majors (or other interested students or community members). The next three talks are aimed at advanced undergraduates or beginning graduate students interested in topology. The last three talks are intended for more advanced graduate students and postdocs.

Schedule

Talks are currently planned to be from 4:30 to 5:30 on (most) Mondays, in Fine 314.

The plan for lectures 5–8 has been updated, thanks to those who filled out the survey. The dates have also been updated to avoid the (similarly named, totally unrelated, exciting) Minerva Lectures.

Abstracts

Here are abstracts for the first few talks; I will write abstracts for later talks once I see how far off schedule I’ve gotten.

Lecture 1. Overview.

This talk is a biased introduction to knot theory and low-dimensional topology, and an overview of the rest of the series. We will start by introducing knotted curves in 3-space and knotted surfaces in 4-space, and discuss how one can represent and (partly) visualize them. We will then discuss different notions of equivalence for knots and surfaces, and assert some theorems and examples related to the existence and uniqueness questions for surfaces. Time permitting, we will introduce the Jones polynomial, an important but still somewhat mysterious tool from the 1980s for studying knots. We will end by overviewing the rest of the series and mentioning some of the most famous (and active) open problems in the area.

Most of this talk should be accessible to undergraduate students who have taken a course that introduces metric spaces.

This lecture (only) will use slides (sorry).

• Supplementary comments and exercises

Lecture 2. Khovanov homology and the $s$-invariant.

This talk is where the work begins. We will start by recalling the Jones polynomial of a knot, then explain its refinement to a chain complex, the Khovanov complex. We will then explain a variant of that construction that gives a number, Rasmussen’s $s$-invariant.

Lecture 3. Alexander polynomial 1 knots are slice.

We will introduce the Alexander polynomial, one of the most famous (and useful) classical invariants of knots, and then prove (modulo one or two hard black-boxed theorems) an important result of Freedman’s, that knots with Alexander polynomial 1 bound topological slice disks.

We will end with a poll of what the audience would like to hear next. In particular, as currently envisioned, the talks focus heavily on the existence question for surfaces; I will ask if the audience would like a detour to uniqueness, or to maintain our focus.

Lecture 4. Four-dimensional topology from Khovanov homology

We will apply Khovanov homology to the existence question for surfaces. We will start by showing the Rasmussen’s invariant obstructs knots from bounding low-genus surfaces, and outline some pain-free proofs that this obstruction is nontrivial. We will then combine these results with Freedman’s theorem from the previous talk (and a few related black-boxed results) to deduce that $\mathbb{R}^4$ admits nonstandard smooth structures.

Lecture 5. The Khovanov stable homotopy type

The Khovanov stable homotopy type is a refinement of Khovanov homology. It takes the form of a CW complex (or CW spectrum) with one cell per generator of the Khovanov complex. There are two constructions: the original one (with Sarkar) in terms of flow categories, and a streamlined version (with Lawson and Sarkar) using an intermediate category of matrices of sets. In this lecture, we will give that streamlined construction (omitting a few proofs).

Lecture 6. Steenrod squares…

Steenrod squares are the simplest interesting example of stable cohomology operations. We will give roughly four constructions of Steenrod squares. This lecture is entirely about classical homotopy theory. It will assume a basic understanding of cohomology, and knowing that cohomology is represented by Eilenberg-MacLane spaces would help.

Lecture 7. …on Khovanov homology

This lecture will combine the previous two to extract some computable additional structure on Khovanov homology: the Steenrod square $\mathrm{Sq}^2: \mathit{Kh}^{i,j}(K;\mathbb{F}_2)\to \mathit{Kh}^{i+2,j}(K;\mathbb{F}_2)$.

Lecture 8. Refinements of the s-invariant

This lecture finishes the series by describing how the operation $\mathrm{Sq}^2$ on Khovanov homology can be combined with the $s$-invariant to obtain a little more information about surfaces in the 4-ball.

Homework

• Lecture 1 Supplementary comments and exercises
• Lecture 2 exercises
• Lecture 3 exercises
• Lecture 4 exercises
• Lecture 5 exercises

References

Here are some references related to the talks. This list is not meant to be comprehensive, and partly reflects where I happened to learn things; I have not even listed all the original papers for results I mentioned.

Links are to arXiv versions, which may not correspond with the published ones. In each section, references listed roughly in the order of the material from references, not in alphabetical order.

Textbooks

Beginning differential topology

• John Milnor, Topology from the Differentiable Viewpoint. Not particularly focused on the classification problems for smooth manifolds discussed in Lecture 1, but a beautiful introduction to smooth topology, via some of its most famous classical theorems.
• Th. Bröcker and K. Jänich, Introduction to Differential Topology. An efficient introduction to abstract smooth manifolds, which I remember finding very helpful when I was first learning the subject.
• Michael Spivak, A Comprehensive Introduction to Differential Geometry. The first two chapters of Volume 1 would mesh nicely with the first lecture. There is also a lot of nice material in the exercises.

John Lee has written more recent, thorough introductions to differential topology and geometry: his Introduction to Topological Manifolds and Introduction to Smooth Manifolds. (I have taught from both and was very happy with them.)

Knot theory

There are lots of introductions to knot theory at many levels, but an excellent one at the level of Lectures 2 - 4 is:

• W. B. Raymond Lickorish, An Introduction to Knot Theory. (This is one of the two math books I brought with me for this visit.)

Non-smooth topology in dimension 4

These aren’t textbooks in the sense of having exercises, but both are written to be readable, and to explain the context in which the new developments occur.

• Michael Freedman and Frank Quinn, Topology of 4-Manifolds.
• Stefan Behrens, Boldizsár Kalmár, Min Hoon Kim, Mark Powell, Arunima Ray, et al, The Disc Embedding Theorem.

Smooth structures in lower dimensions

• Edwin Moise, Geometric Topology in Dimensions 2 and 3. Proves that 3-manifolds admit triangulations.
• William Thurston, Three-dimensional Geometry and Topology. Section 3.10 discusses smoothing triangulations in dimension 3 (results originally due to Whitehead).

Original papers

Khovanov homology and the s-invariant

• Mikhail Khovanov, “A categorification of the Jones polynomial.” Original definition of Khovanov homology. Note that the parameter $c$ in the paper turns out not to give new information; when reading the paper, I suggest setting $c=0$ throughout. (The fact that it does not give new information is proved in his later paper “Link homology and Frobenius extensions.”.)
• Jacob Rasmussen, “Khovanov homology and the slice genus.” Definition of the $s$-invariant and the first combinatorial proof that the torus knot $T_{p,q}$ has slice genus $\frac{(p-1)(q-1)}{2}$. His formulation of the $s$-invariant is somewhat different from the one I gave in lecture.
• Using the deformation from the lecture to define the $s$-invariant is discussed in Section 2 of Robert Lipshitz and Sucharit Sarkar, “A refinement of Rasmussen’s $s$-invariant”. The proof of Theorem 2.2 there is wrong (because we hadn’t actually read what Lee says); the argument we had in mind is given in Proposition 2.1 of our paper “A mixed invariant of non-orientable surfaces in equivariant Khovanov homology”. (Versions of this proof had appeared in the literature many years earlier.) This paper also formulates the deformation as a complex over a polynomial algebra, as I did in the lecture.
• My grading conventions are the same as those in “Khovanov homology of strongly invertible knots and their quotients”. (See Section 2 for a comparison to other parts of the literature.) I believe these also agree with the usual conventions used in the “skein lasagna module” literature; see, for instance, Section 2.3 of Qiuyu Ren and Michael Willis, “Khovanov homology and exotic 4-manifolds”. (Not a surprise: these conventions are the least bad option in several ways, but at the cost of disagreeing with much of the literature.)

Smooth 4-manifolds, topologically slice knots

• Stavros Garoufalidis and Peter Teichner, “On knots with trivial Alexander polynomial”. The relevant part is the appendix.
• Hale Trotter, “On $S$-equivalence of Seifert matrices”.
• James Munkres, “Obstructions to smoothing of piecewise-differentiable homeomorphisms.” Annals of Math. (72), No. 3, 1960, 521-554. Proves that 3-manifolds have smooth structures, building on work of Whitehead and Moise.
• J. H. C. Whitehead, “On $C^1$-complexes.” Annals of Math. (41), No. 4, 1940, 809-824. Discusses smoothing triangulations.
• Edwin E. Moise, “Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung.” Annals of Math. (56), No. 1, 1952, 96-114. Proves that 3-manifolds can be triangulated, and homeomorphisms of 3-manifolds can be approximated by piecewise-linear maps.
• A. V. Černavskiǐ, “Local contractibility of the group of homeomorphisms of a manifold.” Mat. Sb. (N.S.) 79(121), 1969, 307-356. Proves that the group of homeomorphisms of a manifold is locally contractible.

Khovanov homotopy

• Robert Lipshitz and Sucharit Sarkar, “A Khovanov stable homotopy type”. The first construction of our Khovanov stable homotopy type, but uses different techniques from the ones I’m explaining.
• Tyler Lawson, Robert Lipshitz, and Sucharit Sarkar, “Khovanov homotopy type, Burnside category, and products.” Gives the construction of the homotopy type that I’m using in the lectures.
• Po Hu, Daniel Kriz, and Igor Kriz, “Field theories, stable homotopy theory and Khovanov homology”. Mathematically similar to the construction in “Khovanov homotopy type, Burnside category, and products”, though in a somewhat different language.

Expository papers

• Dror Bar-Natan, “On Khovanov’s categorification of the Jones polynomial” is a famous and influential exposition of Khovanov homology. (There have been many others since then.)
• Tyler Lawson, Robert Lipshitz, and Sucharit Sarkar, “The cube and the Burnside category”. Intended as an introduction to the ideas in “Khovanov homotopy type, Burnside category, and products.”
• Robert Lipshitz and Sucharit Sarkar, “Spatial refinements and Khovanov homology”. An overview of the idea of stable homotopy refinements of these kinds of invariants, and a summary of what was known about the Khovanov stable homotopy type.