Topology seminars: 2015-16

I will give a proof that the monoidal category of finite sets and bijections models the sphere spectrum, purely in the language of generalised Reedy model categories.

This is largely an exposition of a well known result of course, but I was motivated by the need of a formulation which helps to understand the relation between dendroidal sets and Gamma spaces.

In this talk, I will introduce a class of accessible model structures on locally presentable categories, which includes, but is more general than, combinatorial model structures. An accessible model structure is particularly good if one wants to left or right induce it along an adjunction - by a theorem of Burke and Garner the induced weak factorisation systems always exist, so one needs to check only a compatibility condition. If it holds then the resulting model structure is again accessible.

One example of an accessible model structure is the Hurewicz model structure on ChR, which can be induced to many categories of interest, like algebras, coalgebras, comodules, comodule algebras, coring comodules and bialgebras.

I will discuss ideas behind some of the proofs for induced model structures and give specific examples. This is joint work with K. Hess, E. Riehl and B. Shipley.

Goodwillie calculus involves the approximation of functors between higher categories by so-called polynomial functors. We show how to associate to a higher category a Goodwillie tower, consisting of categories which are polynomial in an appropriate sense. These approximations enjoy universal properties with respect to polynomial functors.

Furthermore, such Goodwillie towers of higher categories may be classified in terms of the derivatives of the identity functor. This classification can be used to study various localisations of unstable homotopy theory, eg rational homotopy theory, but also "periodic" localisations.

I will describe joint work with Saul Glasman in which we give a completely algebraic construction of the homotopy theory of cyclotomic spectra and the noncommutative syntomic realisation functor. Along the way we prove a conjecture of Kaledin.

A 3-dimensional topological quantum field theory (TQFT) is a representation of the category whose objects are closed surfaces and whose morphisms are 3-dimensional cobordisms.

Recent work has shown that, interestingly, the vector spaces assigned to closed surfaces can be understood graphically in two different ways. Firstly, as a space of string diagrams living on the surface, or secondly, as a space of string diagrams living in a handlebody bounded by the surface.

I will describe the explicit isomorphism between the two pictures. Joint work with Gerrit Goosen.

Coarse geometry is the study of large scale properties of spaces. In coarse geometry, two spaces are considered the same if they "behave the same at infinity", neglecting the fine detail which is important in topology.

For example we consider the real numbers and the integers to be large-scale equivalent, as they look the same when you view them from far away. Although seemingly completely opposite to topology, many results and properties in topology have large-scale analogues in coarse geometry.

In geometric topology, a number of different maps are known as assembly maps, and various conjectures are present which assert that these maps are injective under certain assumptions. The injectivity of these assembly maps give us geometric consequences which are of interest to many.

In this talk, I will introduce the area of coarse geometry and the concept of asymptotic dimension, and explain some of the links between this area and topology. I will give the framework required for a "universal" assembly map for finite asymptotic dimension which will apply to areas such as C*-algebra K-theory, algebraic K-theory and L-theory. A brief discussion of some of these areas will be given as applications for the main result.

Certain 3-dimensional lens spaces are known to smoothly bound 4-manifolds with the rational homology of a ball. These can sometimes be useful in cut-and-paste constructions of interesting (exotic) smooth 4-manifolds. To this end it is interesting to identify 4-manifolds which contain these rational balls.

Khodorovskiy used Kirby calculus to exhibit embeddings of rational balls in certain linear plumbed 4-manifolds, and recently Park-Park-Shin used methods from the minimal model program in 3-dimensional algebraic geometry to generalise Khodorovskiy's result.

The goal of this talk is to give an accessible introduction to the objects mentioned above and also to describe a much easier topological proof of Park-Park-Shin's theorem.

A Picard n-category is a symmetric monoidal n-category in which all cells, including objects, are invertible. The Stable Homotopy Hypothesis states that Picard n-categories should be a model for the homotopy theory of stable n-types. This is known for n=0,1, and in this talk I will discuss some of the challenges moving to the n=2 case.