# Topology Festival

May 10–12, 2019

## Introductory Talks

Maru Sarazola, Cornell University
An introduction to algebraic K-theory

Thomas Ng, Temple University
An introduction to cube complexes and virtual specialness

Qing Liu, Brandeis University
An introduction to hyperbolic boundaries

David Jaz Myers, Johns Hopkins University
How do you identify one thing with another?

## Plenary Talks

### The Winding Invariant and the Andrews-Curtis Conjecture

The Andrews-Curtis Conjecture states that a presentation of the trivial group with $n$ generators and $n$ relators can be transformed into the trivial presentation with relators equal to generators by means of Q*-transformations (moves which modify the relators in a particular way). The conjecture is open even if we allow a larger class of moves called Q**-transformations. A more far-reaching conjecture claims that two presentations of any group are connected by Q**-transformations if their standard complexes are simple homotopy equivalent. In this talk we will give the first example of two presentations which are simple homotopy equivalent and not Q*-equivalent. The relators of these 2-generator 2-relator presentations lie in the commutator subgroup. For each such relator we will construct a curve in the plane whose winding numbers around some points are the coefficients of a Laurent polynomial in two variables. A Q*-transformation translates then into a multiplication by a 2 by 2 matrix with coefficients in the ring of Laurent polynomials, and we will see that a product of those matrices cannot be of the required form.

### Beyond Hyperbolicity: Boundaries of Non-Hyperbolic Spaces

The boundary of hyperbolic $n$-space, $H^n$, has played an important role in the study of hyperbolic manifolds, the dynamics of hyperbolic isometries, and rigidity theorems. Many of these techniques extend to more general Gromov hyperbolic spaces and hyperbolic groups. In this talk we will discuss ways to construct analogous boundaries for non-hyperbolic spaces. Then we will address the question of to what extent these boundaries determine the underlying spaces and the groups that act on them.

### Counterexamples to the Bing-Borsuk Conjecture

A topological space $X$ is topologically homogeneous if, for every two points $x, y\in X$, there is a homeomorphism $h : X \to X$ such that $h(x) = y$. The Bing-Borsuk Conjecture says that every locally compact, locally contractible, finite-dimensional, topologically homogeneous space is a topological manifold. In 1965 Bing and Borsuk proved this in dimensions 1 and 2. John Bryant and the author have disproved this conjecture in dimensions $> 5$, showing in particular that every closed simply connected manifold of dimension $> 5$ has infinitely many topologically distinct homogeneous "fraternal twins" that are counterexamples to Bing-Borsuk.

Whether compact aspherical counterexamples to Bing-Borsuk exist is a very interesting open question.

### Special Covers of Alternating Links

The "virtual conjectures" in low-dimensional topology, stated by Thurston in 1982, postulated that every hyperbolic 3-manifold has finite covers that are Haken and fibered, with large Betti numbers. These conjectures were resolved in 2012 by Agol and Wise, using the machine of special cube complexes. Since that time, many mathematicians have asked how big a cover one needs to take to ensure one of these desired properties.

We begin to give a quantitative answer to this question, in the setting of alternating links in $S^3$. If an alternating link L has a diagram with $n$ crossings, we prove that the complement of $L$ has a special cover of degree less than ($n!$). These special covers have smaller degree than the previously known non-abelian covers of most link complements. As a corollary, we bound the degree of the cover required to get Betti number at least $k$. This is joint work with Edgar Bering.

### Cubical Geometry of Braided Thompson's Group brV

The braided Thompson group brV is a specific subgroup of the mapping class group of the plane minus a Cantor set which is constructed by imitating the definition of Thompson's group V as a group of homeomorphisms of the Cantor space. In this talk, I will explain how to construct an action of brV on an infinite-dimensional $CAT(0)$ cube complex with cube-stabilisers isomorphic to braid groups, and how to deduce that polycyclic subgroups of brV are all virtually free abelian.

### Homotopy Type Theory and Internal Languages of Higher Categories

Homotopy Type Theory (or HoTT) is an approach to foundations of mathematics, building on the homotopy-theoretic interpretation of type theory. In addition to its foundational role, HoTT has been speculated to be the internal language of higher toposes in the sense of Rezk and Lurie. This talk will be an introduction to HoTT, explaining its main ideas and presenting one way in which the connection between type theory and higher categories can be made precise.

### Diffeomorphisms of Disks

What is the homotopy type of the topological group of diffeomorphisms of an $n$-dimensional disk? This is a fundamental question in geometric topology, but its answer remains unknown. I will discuss some partial answers obtained using surgery theory, and how moduli spaces of manifolds and embedding calculus combine to a new strategy to answer to this question.

### Equivariant h-Cobordisms and Algebraic K-Theory

The stable parametrized $h$-cobordism theorem provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold $M$ it gives a decomposition of Waldhausen's $A(M)$ into $QM_+$ and a delooping of the stable $h$-cobordism space of $M$. I will talk about joint work with Malkiewich on this story when $M$ is a smooth compact $G$-manifold.

### Profinite Rigidity

A finitely generated residually finite group $\Gamma$ is called profinitely rigid if whenever $\Lambda$ is a finitely generated residually finite group with $\widehat{\Lambda} \cong \widehat{\Gamma}$ then $\Lambda\cong \Gamma$. In this talk we describe ideas in the proof that establishes that certain (arithmetic) Fuchsian and Kleinian groups are profinitely rigid. These are the first examples of groups that "are like free groups" for which this is known. The case of free groups remains open.

### Quillen's Devissage in Geometry

Abstract: In this talk we discuss a new perspective on Quillen's devissage theorem. Originally, Quillen proved devissage for algebraic $K$-theory of abelian categories. The theorem showed that given a full abelian subcategory $\mathcal{A}$ of an abelian category $\mathcal{B}$, $K(\mathcal{A})\simeq K(\mathcal{B})$ if every object of $\mathcal{B}$ has a finite filtration with quotients lying in $\mathcal{A}$. This allows us, for example, to relate the $K$-theory of torsion $\mathbf{Z}$-modules to the $K$-theories of $\mathbb{F}_p$-modules for all $p$. Generalizations of this theorem to more general contexts for $K$-theory, such as Walhdausen categories, have been notoriously difficult; although some such theorems exist they are generally much more complicated to state and prove than Quillen's original. In this talk we show how to translate Quillen's algebraic approach to a geometric context. This translation allows us to construct a devissage theorem in geometry, and prove it using Quillen's original insights.