In his 1956 paper "On manifolds homeomorphic to the 7-sphere'', John Milnor constructed some examples of manifolds that are homeomorphic, but not diffeomorphic, to the standard unit sphere. They are now called exotic 7-spheres. This example established that the differential structure of a manifold can carry information not given by its topological structure.
In 1986, Thurston introduced a norm on the first cohomology of a 3-manifold $M$ and showed that it can be used to study which cohomology classes are induced by a fibration of $M$ over the circle. In 1998, McMullen introduced a norm on first cohomology that depends only on the Alexander polynomial and showed that it provides a lower bound for the Thurston norm. In this talk, we will introduce the Thurston and Alexander norms and explain why there is an inequality relating the two.
This minicourse provides a friendly, step-by-step introduction to the Kontsevich integral. We begin by demystifying the formula and its construction, showing how it serves as a far-reaching generalization of the classical Gauss linking integral. To establish the invariance of the Kontsevich integral, we explore the holonomy of the Knizhnik–Zamolodchikov (KZ) connection on configuration spaces, utilizing the framework of Chen’s iterated integrals.
This minicourse provides a friendly, step-by-step introduction to the Kontsevich integral. We begin by demystifying the formula and its construction, showing how it serves as a far-reaching generalization of the classical Gauss linking integral. To establish the invariance of the Kontsevich integral, we explore the holonomy of the Knizhnik–Zamolodchikov (KZ) connection on configuration spaces, utilizing the framework of Chen’s iterated integrals.
Modern homology theories have given many knot invariants with the following useful properties: they are additive with respect to connected sum, they give a lower bound for a knot's slice genus, and this lower bound is equal to the slice genus for torus knots. These invariants, called slice-torus invariants, include the Ozsváth–Szabó $\tau$ and Rasmussen $s$ invariants. We discuss how, on a large class of knots, the value of a slice-torus invariant is fully determined by these properties, and can be computed without reference to the homology theory.
One of the fundamental problems in contact topology is to classify contact structures on a given 3-manifold. In particular, classifying contact structures on surgeries along a given knot has been very poorly studied. The only fully understood case so far is that of the unknot (lens spaces); for all other knots we have only partial results, or none at all. Several topological and algebraic tools have been developed to attack this problem.
In this talk, we will discuss the construction of exotic 4-manifolds using Lefschetz fibrations over S^2, which are obtained by finite order cyclic group actions on Σg. We will first apply various cyclic group actions on Σg for g>0, and then extend it diagonally to the product manifolds ΣgxΣg. These will give singular manifolds with cyclic quotient singularities. Then, by resolving the singularities, we will obtain families of Lefschetz fibrations over S^2.
Skein lasagna modules are smooth 4-manifold invariants constructed from functorial link homology theories. These invariants are capable of detecting exotic phenomena in dimension 4. Wall-type stabilization questions ask about the behavior of exotic smooth structures under various topological operations.
Link Floer homology is a powerful invariant of links due to Ozsváth and Szabó. One of its most striking properties is that it detects each link's Thurston norm, a result also due to Ozsváth and Szabó. In this talk I will discuss generalizations of this result to the context of 4-ended tangles, as well as some tangle detection results. This is joint work in progress with Subhankar Dey and Claudius Zibrowius.
