Past Topology Seminar

I will describe the history of the homotopy groups of spheres, and some of the many different roles they have come to play in mathematics.

Consider a diagram of the unknot with c crossings. There is a

sequence of Reidemeister

moves taking this to the trivial diagram. But how many moves are required?

In my talk, I will give

an overview of my recent proof that there is there is an upper bound on the

number of moves, which

is a polynomial function of c.

The question of estimating the length of short conjugators in between
elements in a group could be described as an effective version of the
conjugacy problem. Given a finitely generated group $G$ with word metric
$d$, one can ask whether there is a function $f$ such that two elements
$u,v$ in $G$ are conjugate if and only if there exists a conjugator $g$ such
that $d(1,g) \leq f(d(1,u)+d(1,v))$. We investigate this problem in free
solvable groups, showing that f may be cubic. To do this we use the Magnus
embedding, which allows us to see a free solvable group as a subgroup of a
particular wreath product. This makes it helpful to understand conjugacy
length in wreath products as well as metric properties of the Magnus
embedding.

It's well known that the mapping space of two finite dimensional

manifolds can be given the structure of an infinite dimensional manifold

modelled on Frechet spaces (provided the source is compact). However, it is

not that the charts on the original manifolds give the charts on the mapping

space: it is a little bit more complicated than that. These complications

become important when one extends this construction, either to spaces more

general than manifolds or to properties other than being locally linear.

In this talk, I shall show how to describe the type of property needed to

transport local properties of a space to local properties of its mapping

space. As an application, we shall show that applying the mapping

construction to a regular map is again regular.

Topological field theories give a connection between

topology and algebra. This connection can be exploited in both

directions: using algebra to construct topological invariants, or

using topology to prove algebraic theorems. In this talk, I will

explain an interesting example of the latter phenomena. Radford's

theorem, as generalized by Etingof-Nikshych-Ostrik, says that in a

finite tensor category the quadruple dual functor is easy to

understand. It's somewhat mysterious that the double dual is hard to

understand but the quadruple dual is easy. Using topological field

theory, we show that Radford's theorem is exactly the consequence of

the Dirac belt trick in topology. That is, the double dual

corresponds to the generator of $\pi_1(\mathrm{SO}(3))$ and so the

quadruple dual is trivial in an appropriate sense exactly because

$\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2$. This is part of a large

project, joint with Chris Douglas and Chris Schommer-Pries, to

understand local field theories with values in the 3-category of

tensor categories via the cobordism hypothesis.

I will discuss recent joint work with S. Galatius, in which we

generalise the Madsen--Weiss theorem from the case of surfaces to the

case of manifolds of higher even dimension (except 4). In the simplest

case, we study the topological group $\mathcal{D}_g$ of

diffeomorphisms of the manifold $\#^g S^n \times S^n$ which fix a

disc. We have two main results: firstly, a homology stability

theorem---analogous to Harer's stability theorem for the homology of

mapping class groups---which says that the homology groups

$H_i(B\mathcal{D}_g)$ are independent of $g$ for $2i \leq g-4$.

Secondly, an identification of the stable homology

$H_*(B\mathcal{D}_\infty)$ with the homology of a certain explicitly

described infinite loop space---analogous to the Madsen--Weiss

theorem. Together, these give an explicit calculation of the ring

$H^*(B\mathcal{D}_g;\mathbb{Q})$ in the stable range, as a polynomial

algebra on certain explicitly described generators.

I will talk about the fixed-point properties of matrix groups acting CAT(0) paces, spheres and acyclic manifolds. The matrix groups include general linear groups, sympletic groups, orthogonal groups and classical unitary groups over general rings. We will show that for lower dimensional CAT(0) spaces, the group action of a matrix group always has a global fixed point and that for lower dimensional spheres and acyclic manifolds, a group action by homeomorphisms is always trivial. These results give generalizations of results of Farb concerning Chevalley groups over commutative rings and those of Bridson-Vogtmann, Parwani and Zimmermann concerning the special linear groups SL_{n}(Z) and symplectic groups Sp_{2n}(Z).

This is joint work with Dani Wise and builds on his earlier

work. Let M be a compact oriented irreducible 3-manifold which is neither a

graph manifold nor a hyperbolic manifold. We prove that the fundamental

group of M is virtually special. This means that it virtually embeds in a

right angled Artin group, and is in particular linear over Z.

 The unknotting number of a knot is an incredibly difficult invariant to compute.
In fact, there are many knots which are conjectured to have unknotting number 2 but for
which no proof of this is currently available. It therefore remains an unsolved problem to find an
algorithm that determines whether a knot has unknotting number one. In my talk, I will
show that an analogous problem for links is soluble. We say that a link has splitting number
one if some crossing change turns it into a split link. I will give an algorithm that
determines whether a link has splitting number one. (In the case where the link has
two components, we must make a hypothesis on their linking number.) The proof
that the algorithm works uses sutured manifolds and normal surfaces.

In Riemannian geometry there are several notions of rank

defined for non-positively curved manifolds and with natural extensions

for groups acting on non-positively curved spaces.

The talk shall explain how various notions of rank behave for

mapping class groups of surfaces. This is joint work with J. Behrstock.

For a fixed background manifold $M$ and parameter-space $X$, the associated configuration space is the space of $n$-point subsets of $M$ with parameters drawn from $X$ attached to each point of the subset, topologised in a natural way so that points cannot collide. One can either remember or forget the ordering of the n points in the configuration, so there are ordered and unordered versions of each configuration space.

It is a classical result that the sequence of unordered configuration spaces, as $n$ increases, is homologically stable: for each $k$ the degree-$k$ homology is eventually independent of $n$. However, a simple counterexample shows that this result fails for ordered configuration spaces. So one could ask whether it's possible to remember part of the ordering information and still have homological stability.

The goal of this talk is to explain the ideas behind a positive answer to this question, using 'oriented configuration spaces', in which configurations are equipped with an ordering - up to even permutations - of their points. I will also explain how this case differs from the unordered case: for example the 'rate' at which the homology stabilises is strictly slower for oriented configurations.

If time permits, I will also say something about homological stability with twisted coefficients.

Let G be a finitely generated group generated by g_1,..., g_n. Consider the alphabet A(G) consisting of the symbols g_1,..., g_n and the symbols '+' and '-'. The words in this alphabet represent elements of the integral group ring Z[G]. In the talk we will investigate the computational problem of deciding whether a word in the alphabet A(G) determines a zero-divisor in Z[G]. We will see that a version of the Atiyah conjecture (together with some natural assumptions) imply decidability of the zero-divisor problem; however, we'll also see that in the group (Z/2 \wr Z)^4 the zero-divisor problem is not decidable. The technique which allows one to see the last statement involves "embedding" a Turing machine into a group ring.

The classical associahedra are cell complexes, in fact polytopes,

introduced by Stasheff to parametrize the multivariate operations

naturally occurring on loop spaces of connected spaces.

They form a topological operad $ Ass_\infty $ (which provides a resolution

of the operad $ Ass $ governing spaces-with-associative-multiplication)

and the complexes of cellular chains on the associahedra form a dg

operad governing $A_\infty$-algebras (that is, a resolution of the

operad governing associative algebras).

In classical applications it was not necessary to consider units for

multiplication, or it was assumed units were strict. The introduction

of non-strict units into the picture was considerably harder:

Fukaya-Ono-Oh-Ohta introduced homotopy units for $A_\infty$-algebras in

their work on Lagrangian intersection Floer theory, and equivalent

descriptions of the dg operad for homotopy unital $A_\infty$-algebras

have now been given, for example, by Lyubashenko and by Milles-Hirsch.

In this talk we present the "missing link": a cellular topological

operad $uAss_\infty$ of "unital associahedra", providing a resolution

for the operad governing topological monoids, such that the cellular

chains on $uAss_\infty$ is precisely the dg operad of

Fukaya-Ono-Oh-Ohta.

(joint work with Fernando Muro, arxiv:1110.1959, to appear Forum Math)

I will discuss some aspects of the simplicial theory of

infinity-categories which originates with Boardman and Vogt, and has

recently been developed by Joyal, Lurie and others. The main purpose of

the talk will be to present an extension of this theory which covers

infinity-operads. It is based on a modification of the notion of

simplicial set, called 'dendroidal set'. One of the main results is that

the category of dendroidal sets carries a monoidal Quillen model

structure, in which the fibrant objects are precisely the infinity

operads,and which contains the Joyal model structure for

infinity-categories as a full subcategory.

(The lecture will be mainly based on joint work with Denis-Charles

Cisinski.)

For a fixed n we will investigate homomorphisms Out(F_n) to

Out(F_m) (i.e. free representations) and Out(F_n) to

GL_m(K) (i.e. K-linear representations). We will

completely classify both kinds of representations (at least for suitable

fields K) for a range of values $m$.

Free groups, free abelian groups and fundamental groups of

closed orientable surfaces are the most basic and well-understood

examples of infinite discrete groups. The automorphism groups of

these groups, in contrast, are some of the most complex and intriguing

groups in all of mathematics. In these lectures I will concentrate

on groups of automorphisms of free groups, while drawing analogies

with the general linear group over the integers and surface mapping

class groups. I will explain modern techniques for studying

automorphism groups of free groups, which include a mixture of

topological, algebraic and geometric methods.

Morgan and Culler proved in the 1980’s that a minimal action of a free group on a tree is

completely determined by its length function. This theorem has been of fundamental importance in the

study of automorphisms of free groups. In particular, it gives rise to a compactification of Culler-Vogtmann's

Outer Space. We prove a 2-dimensional analogue of this theorem for right-angled Artin groups acting on

CAT(0) rectangle complexes. (Joint work with M. Margolis)