Past Topology Seminar

Tight contact structures on knot complements arise both from Legendrian realizations of the knot in the standard tight contact structure and from the non-loose Legendrian realizations in the overtwisted structures on the sphere. In this talk, we will deal with negative torus knots. We wish to concentrate on the relations between these various Legendrian realizations of a knot and the contact structures on the surgeries along the knot. In particular, we will build every contact structure by a single Legendrian surgery, and relate the knot properties to the properties of surgeries; namely, tightness, fillability and non-vanishing Heegaard Floer invariant.

Addressing a question of Baker and Reid,

we give a criterion to show that an arithmetic group 

has a congruence subgroup that is algebraically

fibered. Some examples to which the criterion applies

include a hyperbolic 4-manifold group containing infinitely

many Bianchi groups, and a complex hyperbolic surface group.

This is joint work with Matthew Stover.

For a bundle E over a manifold M, the associated "configuration-section spaces" are spaces of configurations of points in M together with a section of E over the complement of the configuration. One often considers subspaces where the behaviour of the section near a configuration point -- a kind of "monodromy" -- is restricted or prescribed. These are examples of "non-local configuration spaces", and may be interpreted physically as moduli spaces of "fields with prescribed singularities" in an ambient manifold.

An important class of examples is given by Hurwitz spaces, which are moduli spaces of branched G-coverings of the 2-disc, and which are homotopy equivalent to certain configuration-section spaces on the 2-disc. Ellenberg, Venkatesh and Westerland proved that, under certain conditions, Hurwitz spaces are (rationally) homologically stable; from this they then deduced an asymptotic version of the Cohen-Lenstra conjecture for function fields, a purely number-theoretical result.

We will discuss another homological stability result for configuration-section spaces, which holds (with integral coefficients) whenever the base manifold M is connected and open. We will also show that the "stabilisation maps" are split-injective (in all degrees) whenever dim(M) is at least 3 and M is either simply-connected or its handle dimension is at most dim(M) - 2.

This represents joint work with Ulrike Tillmann.

The mapping class group of a surface is associated to its Teichmüller space. In turn, its boundary consists of projective measured laminations. Similarly, the group of outer automorphisms of a free group is associated to its Outer space. Now the boundary contains equivalence classes of arationaltrees as a subset. There exist distinct projective measured laminations that have the same underlying geodesic lamination, which is also minimal and filling. Such geodesic laminations are called `non-uniquely ergodic'. I will talk briefly about laminations on surfaces and then present a construction of non-uniquely ergodic phenomenon for arational trees. This is joint work with Mladen Bestvina and Jing Tao.

We will first construct pairs of homotopic 2-spheres smoothly embedded in a 4-manifold that are smoothly equivalent (via an ambient diffeomorphism preserving homology) but not even topologically isotopic. Indeed, these examples show that Gabai's recent "4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis. We will proceed to discuss two distinct ways of obstructing such an isotopy, as well as related invariants which can be used to obstruct an isotopy between pairs of properly embedded disks (rather than spheres) in a 4-manifold.

The square peg problem asks whether every Jordan curve in the
plane contains the vertices of a square. Inspired by Hugelmeyer's approach
for smooth curves, we give a topological proof for "locally 1-Lipschitz"
curves using 4-dimensional topology.

Given two metric spaces $X$ and $Y$, it is natural to ask how faithfully, from the point of view of the metric, one can embed $X$ into $Y$. One way of making this precise is asking whether there exists a coarse embedding of $X$ into $Y$. Positive results are plentiful and diverse, from Assouad's embedding theorem for doubling metric spaces to the elementary fact that any finitely generated subgroup of a finitely generated group is coarsely embedded with respect to word metrics. Moreover, the consequences of admitting a coarse embedding into a sufficiently nice space can be very strong. By contrast, there are few invariants which provide obstructions to coarse embeddings, leaving many seemingly elementary geometric questions open.
I will present new families of invariants which resolve some of these questions. Highlights of the talk include a new algebraic dichotomy for connected unimodular Lie groups, and a method of calculating a lower bound on the conformal dimension of a compact Ahlfors-regular metric space.
 

Witten-Reshetikhin-Turaev quantum invariants of links and 3 dimensional manifolds are obtained from quantum sl(2). There exist different versions of quantum sl(2) leading to other families of invariants. We will briefly overview the original construction and then discuss two variants. First one, so called unrolled quantum sl(2), allows construction of invariants of 3-manifolds involving C* flat connections. In simplest case it recovers Reidemeister torsion. The second one is the non restricted version at a root of unity. It enables construction of invariants of links equipped with a gauge class of SL(2,C) flat connection. This is based respectively on joined work with Costantino, Geer, Patureau and Geer, Patureau, Reshetikhin.

The genus of a knot in a 3-manifold is defined to be the minimum genus of a compact, orientable surface bounding that knot, if such a surface exists. We consider the computational complexity of determining knot genus. Such problems have been studied by several mathematicians; among them are the works of Hass--Lagarias--Pippenger, Agol--Hass--Thurston, Agol and Lackenby. For a fixed 3-manifold the knot genus problem asks, given a knot K and an integer g, whether the genus of K is equal to g. In joint work with Lackenby, we prove that for any fixed, compact, orientable 3-manifold, the knot genus problem lies inNP, answering a question of Agol--Hass--Thurston from 2002. Previously this was known for rational homology 3-spheres by the work of Lackenby.

A broad challenge in the theory of finitely generated groups is to understand their subgroups. A group is commensurably coHopfian if its finite index subgroups are distinct from its infinite index subgroups (that is to say not abstractly isomorphic). We will focus primarily on hyperbolic groups, and give the first examples of one-ended hyperbolic groups that are not commensurably coHopfian.
This is joint work with Emily Stark.
 

In this talk we will survey a novel domain of computational group theory: computing with linear groups over infinite fields.  We will provide an introduction to the area, and will discuss available methods and algorithms. Special consideration is given to algorithms for Zariski dense subgroups. This includes a computer realization of the strong approximation theorem, and algorithms for arithmetic groups. We illustrate applications of our methods to the solution of problems further afield by computer experimentation.

We construct a cellular decomposition of the
Axelrod-Singer-Fulton-MacPherson compactification of the configuration
spaces in the plane, that is compatible with the operad composition.
Cells are indexed by trees with bi-coloured edges, and vertices are labelled by 
cells of the cacti operad. This answers positively a conjecture stated in 
2000 by Kontsevich and Soibelman.

The homological monodromy of the universal family of cubic threefolds defines a representation of a certain Artin-type group into the symplectic group Sp(10;\Z). We use Thurston’s classification of surface automorphisms to prove this does not factor through the genus five mapping class group.  This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds, as established by Clemens and Griffiths.
 

The smooth mapping class group of the 4-sphere is pi_0 of the space of orientation preserving self-diffeomorphisms of S^4. At the moment we have no idea whether this group is trivial or not. Watanabe has shown that higher homotopy groups can be nontrivial. Inspired by Watanabe's constructions, we'll look for interesting self-diffeomorphisms of S^4. Most of the talk will be an outline for a program to find a nice geometric generating set for this mapping class group; a few small steps in the program are actually theorems. The point of finding generators is that if they are explicit enough then you have a hope of either showing that they are all trivial or finding an invariant that is well adapted to obstructing triviality of these generators.

We will introduce the Witt vectors of a ring with coefficients in a bimodule and use them to calculate the components of the Hill-Hopkins-Ravenel norm for cyclic p-groups. This algebraic construction generalizes Hesselholt's Witt vectors for non-commutative rings and Kaledin's polynomial Witt vectors over perfect fields. We will discuss applications to the characteristic polynomial over non-commutative rings and to the Dieudonné determinant. This is all joint work with Krause, Nikolaus and Patchkoria.

I will report on recent joint work with Karen Vogtmann on the Euler characteristic of $Out(F_n)$ and the moduli space of graphs. A similar study has been performed in the seminal 1986 work of Harer and Zagier on the Euler characteristic of the mapping class group and the moduli space of curves. I will review a topological field theory proof, due to Kontsevich, of Harer and Zagier´s result and illustrate how an analogous `renormalized` topological field theory argument can be applied to $Out(F_n)$.

We present a programme towards a combinatorial language for higher (stratified) Morse-Cerf theory. Our starting point will be the interpretation of a Morse function as a constructible bundle (of manifolds) over R^1. Generalising this, we describe a surprising combinatorial classification of constructible bundles on flag foliated R^n (the latter structure of a "flag foliation” is needed for us to capture the notions of "singularities of higher Morse-Cerf functions" independently of differentiable structure). We remark that flag foliations can also be seen to provide a notion of directed topology and in this sense higher Morse-Cerf singularities are closely related to coherences in higher category theory. The main result we will present is the algorithmic decidability of existence of mutual refinements of constructible bundles. Using this result, we discuss how "combinatorial stratified higher Morse-Cerf theory" opens up novel paths to the computational treatment of interesting questions in manifold topology.

The tunnel number of a graph embedded in a 3-dimensional manifold is the fewest number of arcs needed so that the union of the graph with the arcs has handlebody exterior. The behavior of tunnel number with respect to connected sum of knots can vary dramatically, depending on the knots involved. However, a classical theorem of Scharlemann and Schultens says that the tunnel number of a composite knot is at least the number of factors. For theta graphs, trivalent vertex sum is the operation which most closely resembles the connected sum of knots. The analogous theorem of Scharlemann and Schultens no longer holds, however. I will provide a sharp lower bound for the tunnel number of composite theta graphs, using recent work on a new knot invariant which is additive under connected sum and trivalent vertex sum. This is joint work with Maggy Tomova.

I will give a brief survey of some problems in curvature free geometry and sketch

a new proof of the following:

Theorem (Guth). There is some $\delta (n)>0$ such that if $(M^n,g)$ is a closed aspherical Riemannian manifold and $V(R)$ is the volume of the largest ball of radius $R$ in the universal cover of $M$, then $V(R)\geq \delta(n)R^n$ for all $R$.

I will also discuss some recent related questions and results.

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

 For a semisimple modular tensor category the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubachenko-Majid the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. We explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complex valued conformal blocks and a version of factorization and self-sewing via homotopy coends. On the torus we find a derived version of the Verlinde algebra, an algebra over the little disk operad (or more generally a little bundles algebra in the case of equivariant field theories). The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg).

Knots and their groups are a traditional topic of geometric topology. In this talk, I will explain how aspects of the subject can be approached as a homotopy theorist, rephrasing old results and leading to new ones. Part of this reports on joint work with Tyler Lawson.

(joint work with E. Piguet-Nakazawa)

In 2014, Andersen and Kashaev defined an infinite-dimensional TQFT from quantum Teichmüller theory. This Teichmüller TQFT is an invariant of triangulated 3-manifolds, in particular knot complements.

The associated volume conjecture states that the Teichmüller TQFT of an hyperbolic knot complement contains the volume of the knot as a certain asymptotical coefficient, and Andersen-Kashaev proved this conjecture for the first two hyperbolic knots.

In this talk I will present the construction of the Teichmüller TQFT and how we approached this volume conjecture for the infinite family of twist knots, by constructing new geometric triangulations of the knot complements.

No prerequisites in Quantum Topology are needed.

This is a report on work in progress with Jingyin Huang. The complement of an arrangement of linear hyperplanes in a complex vector space has a natural “Borel-Serre bordification” as a smooth manifold with corners. Its universal cover is analogous to the Borel-Serre bordification of an arithmetic lattice acting on a symmetric space as well as to the Harvey bordification of Teichmuller space. In the first case the boundary of this bordification is homotopy equivalent to a spherical building; in the second case it is homotopy equivalent to curve complex of the surface. In the case of a reflection arrangement the boundary of its universal cover is the “curve complex” of the corresponding spherical Artin group. By definition this is the simplicial complex of all conjugates of proper, irreducible, spherical parabolic subgroups in the Artin group. A cohomological method is used to show that the curve complex of a spherical Artin group has the homotopy type of a wedge of spheres.