Topology Festival

We give a basic introduction to the field of 3-manifolds and in particular the role that surfaces play. We cover a fundamental notion of decomposing a 3-manifold, a few important early results including the loop theorem, and mention a recent result resolving the Virtual Haken conjecture. If time permits, we introduce the idea of normal surfaces and its connection to essential surfaces.

The Eckman-Hilton argument arises in topological or algebraic situations where two operations are present which interact well with each other, in a sense. Basic examples are the various ways to concatenate two representatives of the homotopy group \pi_nX of a space X for n>=2, or the two ways to compose higher morphisms in a 2-category. We will make a tour of these examples and the precise conditions we ask from our operations: compatibility, strict vs. weak, etc. In the process we use the argument to get nice results such as the commutativity of \pi_nX.

There are many different ways to think about a torus--we will advocate for "a square with opposite sides identified in a certain way." We will start here and introduce other techniques that are useful when trying to construct examples of spaces of interest to geometric topologists and geometric group theorists, working up to the definition of a hyperbolic orbifold.

For each natural number n, one can consider the set of n-many copies of $\mathbb{N}$, and the group of permutations on this set which are eventual translations: for each permutation, in each copy of $\mathbb{N}$, there is some point beyond which the permutation is just a translation. Brown showed that the Houghton group $H_n$ is of type $F_{n-1}$, but not of type F_{n}. If one replaces "permutation" with "almost surjective injection", one obtains a monoid $M_n$ which serves as vertex set of a cube complex. Lee proved that this complex is CAT(0), and derived an exponential upper bound on the Dehn function of $H_n$ for $n \geq 3$.

In this talk, I will explain the definitions of these objects, showing various useful properties, especially about the cubical structure of the Brown-Lee complex.

A finitely generated, residually finite group G is said to be profinitely rigid if the only finitely generated, residually finite groups with the same set of finite quotients as G are those that are isomorphic to G. More generally, one wants to know which properties P of groups are profinite invariants, i.e. if G has P and H has the same finite quotients as G, does H have P? For example, if G is a 3-manifold group, is H? If G is torsion-free, is H?

I will begin this talk with an overview of how the study of profinite rigidity has thrived in recent years due to a rich interplay between group theory, low-dimensional geometry and arithmetic. I'll then present recent work that underscores the importance of finiteness properties and the homology of groups in this context, describing results that exemplify extremes of rigid and non-rigid phenomena.

In his seminal paper on hyperbolic groups, Gromov tries to extend a beautiful construction of Thurston from dimension 3 to arbitrary n: he builds of a non-positively curved manifold as a branched covering over the n-torus, ramified along codimension-2 subtori. This is particularly interesting in odd dimension n, because every generic fibration of the n-torus lifts to a fibration of the branched cover.

It was later noted by Bestvina that this construction does not yield a negatively-curved space when n>3, due to the presence of 2-flats in the universal cover. In this talk we show that Gromov's construction can be adjusted to produce negatively curved spaces (in fact, hyperbolic manifolds) in at least one specific case in dimension 5. We recover in this way some fibering hyperbolic 5-manifolds constructed with Italiano and Migliorini.

Topological entropy is a fundamental invariant of a dynamical system, measuring its complexity. In this talk, we will discuss connections between the topological entropy of a Hamiltonian system, e.g., a geodesic flow, and the underlying filtered Morse or Floer homology viewed as a persistence module in the spirit of Topological Data Analysis. We will introduce barcode entropy — a Morse/Floer theoretic counterpart of topological entropy — and show that barcode entropy is closely related to topological entropy and that, in low dimensions, these invariants agree. For instance, for a geodesic flow on any closed surface the barcode entropy is equal to the topological entropy. The talk is based on joint work with Erman Cineli, Basak Gurel and Marco Mazzucchelli.

The Eckmann–Hilton argument gives us circumstances in which commutativity arises from having two binary operations that interact in certain coherent ways. The argument is used to show that all higher homotopy groups are Abelian, but can be stated in generality as a piece of algebra: if a set is equipped with two monoid structures satisfying a type of interchange, then the two binary operations coincide and are commutative.

The Eckmann–Hilton argument arises in higher-dimensional category theory via degeneracy, that is, higher-dimensional structures where some of the lowest dimensions are trivial. More generally, commutativities arise when we consider cells with identity boundaries, as the cells can then commute past each other via interchange. Degenerate structures become key test cases for theories of higher category, enabling us to examine whether the appropriately weak forms of commutativity arise. For sets there is only strict commutavitiy, but for categories there is space for a weak form of commutativity called a braiding, and a stricter one called a symmetry. For 2-categories there is another possibility in between, and as we add in dimensions there are more and more nuanced possibilities for flavors of commutativity. These are arranged into the Periodic Table of Higher Categories, conjectured by Baez and Dolan. As we increase dimensions we also need to generalise the Eckmann–Hilton argument into higher dimensions, with more binary operations, and more interchange laws.

In this talk we will give an overview of the basic case of doubly-degenerate 2-categories, before moving on to the higher-dimensional argument showing how doubly-degenerate 3-categories give braided monoidal categories, as in my recent work with Alexander S. Corner. We will also introduce work in progress on $n$-degenerate $(n+1)$-categories.

Grothendieck’s Homotopy Hypothesis states that homotopy n-types are modeled by n-groupoids, and by extension, that spaces are modeled by $\infty$-groupoids. Moreover, this equivalence is induced by the homotopy groups of both constructions.

While there are some geometric solutions to the Hypothesis, the most interesting solution would be to show that some completely algebraic structures are equivalent to topological ones. Unfortunately, the Homotopy Hypothesis is known to be false for strict higher groupoids, and fully weak ones are too complex to work with.

In this talk, I will present an operadic method for calculating the homotopy groups of algebraic higher groupoids. I will then use this to identity models algebraic of n-groupoid that are tractable enough to work with directly, but which escape the homotopy degeneracies that appear in the strict setting.

Symplectic geometry is a generalization of classical mechanics, in which position and momentum coordinates are paired. Mathematically, a symplectic manifold is an even-dimensional manifold carrying a "symplectic form" – a closed, nondegenerate 2-form. In two dimensions, symplectic geometry is equivalent to volume-preserving geometry, but in higher dimensions, Gromov proved in 1985 that an embedding from a finite-volume ball into an infinite-volume cylinder can only preserve the symplectic form if the ball embeds via the identity. Symplectic geometers have studied generalizations of Gromov's result ever since. In this talk, we will show that in many four-dimensional examples, requiring the complement of the embedding to contain a symplectic surface with fixed boundary conditions (the so-called "anchor") provides an even stronger restriction than the symplectic form alone. Our examples have a toric structure, and when symplectic embeddings between them are anchored we show they must also be toric. The main tool is the interplay between the action filtration and intersection number in embedded contact homology, which we will review. Joint work with Michael Hutchings, Agniva Roy, and Yuan Yao.

Knot Floer homology is a powerful invariant of knots and links, developed by Ozsvath and Szabo in the early 2000s. Among other properties, it detects the genus, detects fiberedness, and gives a lower bound to the 4-ball genus. The original definition involves counting homomorphic curves in a high-dimensional manifold, and as a result the invariant can be hard to compute. In 2007, Manolecu, Ozsvath, and Sarkar came up with a purely combinatorial description of knot Floer homology for knots in the 3-sphere, called grid homology. Soon after, Ozsvath, Szabo, and Thurston defined invariants of Legendrian knots using grid homology. We show that the filtered version of these GRID invariants, and consequently their associated invariants in a certain spectral sequence for grid homology, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure, strengthening a result of Baldwin, Lidman, and Wong. This is joint work with Jubeir, Schwartz, Winkeler, and Wong.

Artin groups are a family of groups that generalize braid groups, and can be defined and studied from various perspectives: topologically, algebraically, and combinatorially. They are mysterious - a lot of basic questions about them remain unanswered. Artin groups are given by simple looking group presentations, are closely related to Coxeter groups, and arise as the fundamental groups of certain complex hyperplane arrangements. In this talk, I will focus on profinite properties of Artin groups. Informally, those properties tell us if the geometry and algebra of the group can be approximated by its finite quotients. I will mention some joint work with Kevin Schreve.

Suppose that a finitely presented group H is a normal subgroup of a hyperbolic group with free quotient. By a result of Gersten and Short the group H has a polynomial isoperimetric inequality. This is established by starting with a geometrically controlled disk filling in the ambient hyperbolic group of a loop in H and then pushing this filling into H. We describe other situations where one can push fillings with respect to height functions on spaces and give applications to subgroups of right-angled Artin groups and to Houghton groups.

Counting embedded curves on a surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting surfaces in a 3-manifold, with the key difference that now the surfaces themselves have more intrinsic topology. As there are only finitely many essential surfaces of bounded Euler characteristic up to isotopy in an atoroidal 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Haken’s normal surface theory, we can characterize not just the rate of growth but show the exact count is a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples.

This is joint work with Stavros Garoufalidis and Hyam Rubinstein and the reference is our paper with the same title. The only background I will assume is the notion of a manifold, the genus of a surface, and a little about the fundamental group.