Topology seminars: 2017-18

In the last few years, two different resolutions of the K(2)-local sphere at the prime 3 have been used very successfully to settle some basic problems in K(2)-local stable homotopy theory like the chromatic splitting conjecture, the calculation of Hopkins' K(2)-local Picard group and determining $K(2)-local Brown-Comentz duality. The focus is now moving towards the prime 2 where one can hope for similar progress.

In this talk we concentrate on one of these two resolutions, the centraliser resolution at the prime 2.

Two dimensional conformal field theories (CFTs) are conformally invariant quantum field theories on a two dimensional manifold. What distinguishes two dimensions from all others is that the (Lie) algebra of local conformal transformations become infinite dimensional. This extraordinary amount of symmetry allows certain conformal field theories to be solved by symmetry considerations alone.

The most intensely studied type of CFT, called a rational CFT, is characterised by the fact that its representation theory is completely reducible and that there are only a finite number isomorphism classes of irreducibles. The representation categories of these CFTs form so called modular tensor categories which have important applications in the construction of 3-manifold invariants.

In this talk I will discuss recent attempts at generalising this very rich structure to CFTs whose representation categories are neither completely reducible nor finite.

In this talk I will explain a way to detect groups in the geometric isotropy of a compact rational global spectrum. As an application, I will show that the Balmer spectrum of the rational global stable homotopy category exhibits at least two different types of prime: group and multiplicative primes.

I will introduce the notion of rational cohomological dimension of topological spaces and show a simple way to calculate it when we restrict ourselves to a certain class of topological spaces.

Very roughly, the r.c.d of a space X is the largest p such that the pth rational cohomology of X is non-zero. This invariant can be calculated in terms of the more geometric notion of sheaves on X.

The category of sheaves on X is an abelian category and the injective dimension of this category is the r.c.d of X. This is a standard way to calculate the the r.c.d. of a space, but can be rather difficult.

In this talk, I will describe how for profinite spaces, this injective dimension is related to a simpler notion: the Cantor-Bendixson dimension of the space. There will be a number of pictures and some nice examples illustrating the calculations.

Enumerative geometry studies questions of the type: how many geometric objects satisfy a prescribed set of (generic) conditions? Over the complex field the answer is a single number. However, over R the answer depends on the configuration.

A theorem of Borel and Haefliger states that mod 2 the answer is the same. Thom realised, that for a generic a) smooth, b) holomorphic map f, the cohomology class [Si(f)] of the singular points of f of a given type can be expressed as a universal polynomial evaluated at the characteristic classes of the map.

The second theorem of Borel and Haefliger states that mod 2, the universal polynomial is the same in the smooth and holomorphic case.

In this talk I plan to discuss these questions from the point of view of equivariant topology. The spaces satisfying the condition of the Borel-Haefliger theorem are part of a class of Z2-spaces called conjugation spaces introduced by Hausmann, Holm and Puppe. Analogously we introduce a class of U(1)-spaces which we call circle spaces in an attempt to say something more than parity about these questions. This is joint work with László Fehér.

We consider the problem of lifting certain Quillen model structures on the category of cyclic sets to the category of groupoids, echoing the construction of the Thomason model structure on Cat. We prove that this model structure only captures the theory of homotopy 1-types, and as a consequence, that SO(2)-equivariant homotopy 1-types cannot be encoded in a discrete manner.

We will fully describe all of the components required for this model structure, in particular, assuming no familiarity with the model structures on cyclic sets or the Thomason model structure on Cat. This work is joint with Richard Garner.

Maurer-Cartan elements for differential graded Lie algebras or associative algebras play an important role in several branches of mathematics, in particular for classifying deformations. There are different sensible notions of equivalence for Maurer-Cartan elements, and while they agree in the nilpotent case, the general theory is not yet well-understood.

This talk will compare gauge equivalence and different notions of homotopy equivalence for Maurer-Cartan elements of a dg-algebra. As an application we extend the study of cohesive modules introduced by Block, and find a new algebraic characterisation of infinity local systems on a topological space. This is joint work with Joe Chuang and Andrey Lazarev.

Equivariant stable homotopy concerns the study of objects with symmetry. It has been shown recently by Patchkoria that the G-equivariant stable homotopy category is uniquely determined by its triangulated structure, G-action and induction/ transfer/ restriction maps.

In particular this implies that all reasonable categories of G-spectra realise the same homotopy theory. We consider this result with respect to equivariant K-theory, which merges model category techniques, equivariant structures and calculations from the stable homotopy groups of spheres.

One of the basic tools to study a tensor-triangulated category is a classification of its thick tensor ideals. In my talk, I will discuss such a classification for the category of compact G-spectra for a finite abelian group G. This is joint work with Tobias Barthel, Niko Naumann, Thomas Nikolaus, Justin Noel and Nat Stapleton, and builds on work of Strickland and Balmer-Sanders.

Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories provided a beautiful link between the geometry of "spacetimes" (cobordisms) and algebraic structures. Combining this with the physical notion of "locality" led to the introduction of the language of higher categories into the topic. Natural targets for extended topological field theories are higher Morita categories: generalisations of the bicategory of algebras, bimodules, and homomorphisms.

After giving an introduction to topological field theories, I will explain how one can use geometric arguments to obtain results on dualisablity in a "factorisation version" of the Morita category and using this, examples of low-dimensional field theories "relative" to their observables. An example will be given by polynomial differential operators, ie the Weyl algebra, in positive characteristic and its centre. This is joint work with Owen Gwilliam.

Rational Mackey functors for a compact topological group G are a useful tool for modelling rational G equivariant cohomology theories. Having a better understanding of Mackey functors will enhance our understanding of G-cohomology theories and G-equivariant homotopy theory in general.

In the compact Lie group case, rational Mackey functors have been studied extensively by John Greenlees (and others). In this talk we will discuss what can be shown in the case where G is profinite (an inverse limit of finite groups).