Northeastern University
Weiming Lake, Peking University
Tsinghua University

Workshop on Interactions of 3- & 4-dimensional Topology

March 10-12, 2023 (Beijing time)
Home Registration Program Participants

With the speakers' permission, we posted some of the recordings here.

Friday, March 10, 2023 (Beijing time)

8:25 am Opening Remarks & Group Shot!
8:30 am - 9:30 am Skein lasagna modules
Ciprian Manolescu (Stanford University)
Abstract: Skein lasagna modules were introduced by Morrison, Walker and Wedrich as an extension of Khovanov-Rozansky homology to the setting of a four-manifold and a link in its boundary. I will describe how to express the skein lasagna module in terms of a handle decomposition for the four-manifold, and compute some examples. The talk is based on joint work with Ikshu Neithalath, and on joint work with Kevin Walker and Paul Wedrich.
9:40 am - 10:40 am Lattice cohomology and q-series invariants of plumbed 3-manifolds
Slava Krushkal (University of Virginia)
Abstract: I will introduce a combinatorially defined invariant of negative definite plumbed 3-manifolds, equipped with a spin-c structure. It unifies and extends two theories with rather different origins and structures. One is the theory of lattice cohomology, closely related to Heegaard Floer homology. The other invariant of 3-manifolds, with origins in physics, is a certain q-series defined by Gukov-Pei-Putrov-Vafa, which is conjectured to recover quantum invariants of 3-manifolds at roots of unity. (Joint work with R. Akhmechet and P. Johnson)
10:50 am - 11:50 am Slice obstructions from genus bounds in definite 4-manifolds
Maggie Miller (Stanford University)
Abstract: Minimum genus bounds for surfaces representing specific homology classes in some 4-manifolds (especially connected sums of \(CP^2\)s) can be used to show that certain knots in \(S^3\) are not slice. For example, one genus bound due to Bryan in the 1990s can be used to show that the (2,1)-cable of the figure eight knot is not slice, recovering a result of Dai-Kang-Mallick-Park-Stoffregen from last summer based on Heegaard Floer (and related) homology. I'll talk about this construction, underlying motivation, and some interesting open questions that follow. This is joint work with Paolo Aceto, Nickolas A. Castro, JungHwan Park and András Stipsicz.
Noon Lunch
1:50 pm - 2:50 pm Theta-graph surgery for diffeomorphisms of 4-manifolds
Tadayuki Watanabe (Kyoto University)
Abstract: We constructed diffeomorphisms of 4-manifolds by surgery along theta-graphs embedded in 4-manifolds. In this talk, I will talk about our results on some properties of theta-graph surgery: nontriviality up to isotopy, extension to pseudo-isotopy, fiberwise positive scalar curvature metrics. To obtain some of such properties, we describe theta-graph surgery by families of framed links of 1- and 2-spheres in 4-manifolds and modify them by an analogue of Kirby calculus. To prove the nontriviality for some 4-manifolds, we use an analogue of Kontsevich's configuration space integral invariant for a two-loop graph, which is in a sense a higher dimensional analogue of the Casson invariant, according to a result by G. Kuperberg, D. Thurston, J. Marché, and C. Lescop.

Part of this work is joint with Boris Botvinnik (partially based on a suggestion by Peter Teichner).
3:00 pm - 4:00 pm One-domination of 4-manifolds with cyclic fundamental groups
Yang Su (Chinese Academy of Sciences)
Abstract: Based on the topological classifications of 4-manifolds with cyclic fundamental groups, we study the existence and finiteness of degree one maps between these manifolds. This is a joint work with Shicheng Wang and Zhongzi Wang.
4:00 pm - 4:20 pm Coffee Break
4:20 pm - 5:20 pm Floer homology and non-fibered knot detection
Steven Sivek (Imperial College)
Abstract: Knot Floer homology and Khovanov homology are known to detect a small handful of knots, including the unknot, the trefoils, the figure eight, and the cinquefoil. In each case except that of the unknot, the fiberedness of the knots in question plays a crucial role in the proofs.  In this talk, I'll discuss recent work with John Baldwin in which we show for the first time that both invariants can also detect non-fibered knots, including \(5_2\), and that HOMFLY homology detects infinitely many knots. The key input is a classification of genus-1 knots which are "nearly fibered" from the perspective of knot Floer homology.
5:30 pm - 6:30 pm Lecture 1
Mark Powell (University of Glasgow)
Abstract: Freedman's disc embedding theorem allows us to perform the Whitney trick in dimension 4, under special circumstances. It underpins all of topological 4-manifold theory. In the first lecture, I will explain an outline of the proof.

Saturday, March 11, 2023 (Beijing time)

8:30 am - 9:30 am Groups of automorphisms of hyperbolic manifolds
Ryan Budney (University of Victoria)
Abstract: Given a compact hyperbolic manifold of dimension \(n\geq 2\) its isometry group is known to be finite. In dimensions \(n\geq 3\), Mostow Rigidity tells us the space of homotopy self-equivalences of the manifold has the isometry group as a deformation-retract. In dimension 3, due to work of Hatcher, Waldhausen and Gabai, we know the group of homeomorphisms, PL-automorphisms and diffeomorphisms also has the isometry group as a deformation-retract. In dimensions \(\geq 11\), Farrell and Jones used the machinery of Higher Simple Homotopy Theory (K-theory, pseudoisotopy, etc) to show that smooth, PL and topological automorphism groups of compact hyperbolic manifolds do not have the homotopy-type of the isometry group, moreover they do not have the homotopy-type of finite CW-complexes. We extend these results to dimension \(n\geq 4\) by avoiding the use of Higher Simple Homotopy Theory, and constructing diffeomorphisms explicitly using "barbells".
9:40 am - 10:40 am An atomic approach to Wall-type stabilization problems
Kyle Hayden (Rutgers University)
Abstract: Wall-type stabilization problems concern the collapse of exotic 4-dimensional phenomena under stabilization operations (such as taking connected sums with \(S^2\times S^2\). I will describe an elementary approach to producing exotic 4-manifolds and knotted surfaces that are candidates to remain exotic after stabilization. As a proof of concept, I will use this to present examples of exotically knotted surfaces in the 4-ball that remain exotic after (internal) stabilization, detected by the cobordism maps on universal Khovanov homology.
10:50 am - 11:50 am Instanton Floer homology and Heegaard diagrams
Zhenkun Li (Stanford University)
Abstract: Instanton Floer homology was introduced by Floer in 1980s and it has become a power invariant for three manifolds and knots since then. It has lead to many milestone results, such as the approval of Property P conjecture. Heegaard diagrams, on the other hand, is a combinatorial method to describe and study 3-manifolds. In principle, a Heegaard diagram determines a 3-manifold and hence determines its instanton Floer homology as well. However, no explicit relations between Heegaard diagrams and instanton Floer homology were known until a recent progress by my collaborators and I. In this talk, for a 3-manifold \(Y\), I will present how to extract some information about the instanton Floer homology of \(Y\) from Heegaard diagrams of \(Y\). I will also discuss on some applications and possible future directions. This is a joint work with Baldwin and Ye.
Noon Lunch
1:50 pm - 2:50 pm \(gl(1|1)\)-Alexander polynomial for 3-manifolds
Yuanyuan Bao (University of Tokyo)
Abstract: The Alexander polynomial introduced by Viro based on the representations of \(U_q(gl(1|1))\) has many interesting connections with Heegaard Floer homology. For instance, A. Manion showed that the decategorification of Ozsvath and Szabo's bordered theory coincides with the functor given by Viro. From a different viewpoint, Zhongtao Wu and I studied the Alexander polynomial for MOY graphs (framed trivalent graph with coloring), which is in fact the Euler characteristic of the Heegaard Floer homology of graphs. We later proved that this polynomial is equivalent to Viro's \(U_q(gl(1|1))\)-polynomial. In this talk, we show how to construct a 3-manifold invariant from Viro's \(U_q(gl(1|1))\)-polynomial, by following the method proposed by Castantino, Geer and Patureau-Mirand. This result is joint work with Noboru Ito.
3:00 pm - 4:00 pm Spaces of embeddings of 3-manifolds in 4-manifolds
Daniel Ruberman (Brandeis University)
Abstract: There are many interesting obstructions to the existence of smooth embeddings of a surface or 3-manifold in a given 4-manifold, and examples of embeddings that are not smoothly isotopic. Often the topological version of existence and uniqueness questions have different answers from the smooth case. In joint work with Dave Auckly, we compare the spaces of smooth and topological embeddings. We show that for many 3-manifolds, there is a large kernel of the map on the homology and homotopy groups of the space of smooth embeddings to the corresponding space of topological embeddings.
4:00 pm - 4:20 pm Coffee Break
4:20 pm - 5:20 pm The (2,1)-cable of the figure-eight knot is not smoothly slice
Sungkyung Kang (IBS Center for Geometry and Physics)
Abstract: We prove that the (2,1)-cable of the figure-eight knot is not smoothly slice. This answers a question of Kawauchi, posed originally in 1980.
5:30 pm - 6:30 pm Lecture 2
Mark Powell (University of Glasgow)
Abstract: Kreck's modified surgery theory is a powerful method for understanding manifolds and their symmetries.  In this lecture I will introduce this method, and explain how to use it to classify 4-manifolds up to stable diffeomorphism, giving detailed examples. I will finish by comparing stable diffeomorphism with a range of popular equivalence relations on 4-manifolds, again giving many examples.

Sunday, March 12, 2023 (Beijing time)

8:30 am - 9:30 am Floer homology and right-veering surface diffeomorphisms
John Baldwin (Boston College)
Abstract: A diffeomorphism of a surface with boundary is said to be right-veering if it sends every properly embedded arc in the surface to the right near the boundary. This dynamical notion is intimately related with contact geometry, as shown by Honda-Kazez-Matic. In this talk I'll explain how knot Floer homology can be used to completely detect whether a surface diffeomorphism is right-veering. This gives a purely Floer-theoretic characterization of tight contact structures, and has applications to Dehn surgery and taut foliations. Our proof uses a relationship between Heegaard Floer homology and the periodic Floer homology of surface diffeomorphisms. This is based on joint work with Yi Ni and Steven Sivek.
9:40 am - 10:40 am Geometry of positive symplectic rational surfaces
Tian-jun Li (Unversity of Minnesota)
Abstract: A symplectic rational surface is called positive if the pairing between the two degree 2 cohomological invariants, the symplectic class and the first Chern class, is positive. A symplectic rational surface with Euler number up to 12 is positive and a symplectic toric surface is positive. We will discuss some recent results on the geometry of such surfaces. This is based on joint works with Jun Li, Weiwei Wu, and with Jie Min and Shengzhen Ning.
10:50 am - 11:50 am Introduction to Ricci Flow
Bennett Chow (University of California, San Diego)
Abstract: In this talk I will give an introduction to Ricci flow for non-specialists. A lengthier introduction will be given via Tencent in Short course on "A Retrospective Look at Ricci Flow".
Noon Lunch
1:50 pm - 2:50 pm Reverse Lagrangian surgery on fillings
Yu Pan (Tianjin University)
Abstract: For an immersed filling of a topological knot, one can do surgery to resolve a double point with the price of increasing surface genus by 1. In the Lagrangian analog, one can do Lagrangian surgery on immersed Lagrangian fillings to treat a double point by a genus. In this talk, we will explore the possibility of reversing the Lagrangian surgery, i.e., compressing a genus into a double point. It turns out that not all Lagrangian surgery are reversible.
3:00 pm - 4:00 pm A new approach to light bulb tricks
Danica Kosanović (Swiss Federal Institute of Technology in Zürich)
Abstract: I will explain a new approach to geometric dual spheres in 4-manifolds, which can not only classify isotopy classes of disks or spheres which have such a dual, but also leads to some new insights into mapping class groups of 4-manifolds. This is joint work with Peter Teichner.
4:00 pm - 4:20 pm Coffee Break
4:20 pm - 5:20 pm On spaces of smooth structures on 4-manifolds
Jianfeng Lin (Tsinghua University)
Abstract: Given a manifold \(X\), the space of smooth structure on \(X\) (denoted by \(Sm(X)\)) is defined to be the homotopy quotient of the homeomorphism group by the action of diffeomorphism group. This space measures the difference between the topology of these two groups. In particular, \(Sm(X)\) is always contractible when \(X\) is 3-dimensional or less. In this talk, I will sketch a proof that \(Sm(X)\) has a nontrivial rational homotopy group when \(X\) is any compact orientable 4-manifold \(X\) (with or without boundary) or is the 4-dimensional Euclidean space. This is a joint work with Yi Xie. 
5:30 pm - 6:30 pm Lecture 3
Mark Powell (University of Glasgow)
Abstract: I will explain how to use modified surgery together with the disc embedding theorem to obtain homeomorphism classifications. As an extended example, I will outline the modified surgery proof of the classification of 1-connected closed 4-manifolds.  I will also give an application to isotopy of embedded surfaces.