Topology seminars: 2016-17

Rational G-equivariant cohomology theories can be classified in the sense that there is an algebraic model for them. The model can be viewed as a category of sheaves over the space of subgroups of G. This has the character of a category of sheaves of modules over an algebraic variety we might call the spectrum of the sphere.

The notion of a derived A-infinity algebra, due by Sagave, is a generalisation of the classical A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field.

I will describe a hierarchy of notions of homotopy between the morphisms of such algebras, in such a way that r-homotopy equivalences underlie E_r-quasi-isomorphisms, defined via an associated spectral sequence.

Along the way, I'll give two new interpretations of derived A-infinity algebras. This is joint work with Joana Cirici, Daniela Egas Santander and Muriel Livernet.

Gorenstein duality is a homotopy theoretic framework that allows one to view a number of dualities in algebra, geometry and topology as examples of a single phenomenon.

I will briefly introduce the framework and concentrate on illustrating it with examples coming from derived algebraic geometry, especially topological modular forms with level structure.

In equivariant homotopy theory there are some theories that are defined in a uniform way for all groups in a specific class, rather than just for a particular group. The idea of global stable homotopy theory is to view this collection of compatible equivariant theories as one "global" object.

One way to formalise this idea is to consider the well-known category of orthogonal spectra and to use a finer notion of equivalence: the global equivalence.

In this talk, I will give an overview on global stable homotopy theory via orthogonal spectra and I will present a global equivariant version of connective topological K-theory. Time permitting, I will explain how to generalise this construction to obtain a global equivariant version of connective K-theory of C*-algebras.

The Bousfield semiring controls many interesting phenomena in stable homotopy theory. The literature contains many fragmentary results about the structure of this semiring. I will report on a project to combine all of these results into a single consolidated statement.

This seems the right time of the year in Sheffield to talk about the functor which associates to a tree its set of leaves. It is a functor form the indexing category Omega for dendroidal sets to the indexing category Gamma for Gamma-spaces. Gamma-spaces form a classical model for E-infinity algebras and infinite loop spaces.

We will show that this leaves functor induces an equivalence of homotopy categories for suitable Quillen model structures on dendroidal sets.

At the prime 2, there are 4 Hopf invariant one elements (mod 2). These can be used to build some small complexes which also appear as low dimensional skeleta of some important classifying spaces and Thom spectra over them.

Passing to free infinite loop spaces we can build some additional Thom spectra E-infinity ring spectra which have interesting properties. These have E-infinity ring maps to some important spectra including kO and tmf.

I will describe these spectra and some conjectures about splitting them and survey what is known so far.

In this talk, the following topics in algebraic topology will be briefly outlined:

• Strong (co)homology groups

• Phantom maps

• Comultiplications on a wedge of spheres

• The same n-type structures of CW-complexes

I will report on joint work with Lucio Cirio on categorifications of the Lie algebra of chord diagrams via infinitesimal 2-braidings in differential crossed modules.

Configuration spaces appear in many areas of mathematics. They are simple to define but produce extremely complicated spaces.

In this talk, we will introduce a family of posets, indexed by the natural numbers and finite sets, called the poset of chained linear preorders. It turns out that the geometric realisations of these posets are homotopy equivalent to configuration spaces on real vector spaces, and that the combinatorics involved can reveal some of the homotopical properties of these spaces.

In the last two decades applied topologists have developed numerous methods for ‘measuring’ and building combinatorial representations of the shape of the data. The most famous example of the former is persistent homology and of the latter, mapper.

I will briefly talk about both of these methods and show several successful applications. Time permitting, I will talk about my work on making persistent homology easier to combine with standard machine learning tools.

An equivariant infinite loop space machine is a functor that constructs genuine equivariant spectra out of simpler categorical or space level data. In the late 80's Lewis-May-Steinberger and Shimakawa developed generalisations of the operadic approach and the Gamma-space approach respectively.

In this talk, I will describe work in progress that aims to understand these machines conceptually, relate them to each other, and develop new machines that are more suitable for certain kinds of input. This work is joint with Anna Marie Bohmann, Bert Guillou, Peter May and Mona Merling.

The Balmer spectrum of the category of rational G-spectra as a poset is the closed subgroups of G under cotoral inclusion. In December, I posted a preprint on the arXiv that proved this for tori: the talk will describe a much simpler proof of a theorem for all compact Lie groups.

Bar constructions appear in topology and algebra as a tool to produce good resolutions. While studying distributive laws for homotopy coherent monads, we began thinking about using the bar construction to produce cofibrant replacements for certain objects in the Joyal model structure. This is a standard approach in homotopy theory, but we could not find results in the literature that were general enough to cover our application.

In this talk, I will try to explain a little about the problem we wanted to solve related to distributive laws and then go on to give a general result about bar constructions in enriched model categories. This is joint work with Daniel Schaeppi.

Graded E-theory is a bivariant functor from the category where objects are graded C*-algebras and arrows are graded *-homomorphisms to the category where objects are abelian groups and arrows are group homomorphisms. It is bivariant in the sense that it is a cohomology theory in its first variable and a homology theory in its second variable.

In this talk I'll give a description of a quasi-topological space and explain why this notion is necessary in our case. We will define the notion of an orthogonal quasi-spectrum as an orthogonal spectrum for quasi-topological spaces, and further give the quasi-topological spaces to form the spectrum for graded E-theory. If time allows I will give the smash product structure.

Recently, there has been some new understanding of various possible commutative ring G-spectra. In this talk I will recall these possibilities and discuss the most naive (or trivial) commutative ring G-spectra.

Then I will sketch the main ingredients coming into the proof that if G is finite and we work rationally these objects correspond to (the usual) commutative differential algebras in the algebraic model for rational G-spectra. This is joint work with David Barnes and John Greenlees.

The Balmer spectrum of the category of rational G-spectra as a poset is the closed subgroups of G under cotoral inclusion. In December, I posted a preprint on the arXiv that proved this for tori: the talk will describe a much simpler proof of a theorem for all compact Lie groups.

The method applies in other contexts with only a few special inputs from equivariant topology: the Localisation Theorem, The calculation of the Burnside ring and a method of calculation for maps between free G-spectra.

Homology groups provide bounds on the minimal number of handles needed in any handle decomposition of a manifold. We will use Casson-Gordon invariants to get better bounds in the case of 4-dimensional rational homology balls whose boundary is a given rational homology 3-sphere. This analysis can be used to understand the complexity of the discs associated to ribbon knots in S^3. This is a joint work with P. Aceto and M. Golla.

Cohomology operations are a very useful property of a cohomology theory. The collection of cohomology operations has a very rich structure. Historically the dual notion, of homology cooperations, have been the main target of attention and a nice algebraic structure called a Hopf ring has been used to understand these.

Unfortunately, the Hopf ring contains no structure that is dual to the notion of composition. Boardman, Wilson and Johnson attempt to rectify this situation by defining an enriched Hopf ring, although this structure is rather less pleasant.

A 2009 theorem of Stacey and Whitehouse shows that the collection of cohomology operations has the structure of an algebraic object called a plethory and this expresses all the structure, including composition. In this talk I shall define the above concepts and illustrate some examples of plethories for known cohomology theories.

Weighted projective spaces are interesting through many lenses: for example, as natural generalisations of ordinary projective spaces, as toric varieties and as orbifolds. From the point of view of algebraic topology, it is natural to study their algebraic topological invariants – notably, their (equivariant) cohomology rings.

Recent work has provided satisfying qualitative descriptions for these rings, in terms of piecewise algebra, for various cohomology theories.

This talk will introduce weighted projective spaces as toric varieties and survey results on their (equivariant) cohomology rings, with particular focus on equivariant K-theory. It will conclude with recent results of Megumi Harada, Tara Holm, Nige Ray and the speaker, and indicate the flavour of current work of Tara Holm and the speaker.

If O is an operad (in a friendly category, e.g. the category of S-modules of stable homotopy theory), M is an O-bimodule, A is an O-algebra, then the circle product over O of M with A is again an O-algebra. A useful derived version is the bar construction B(M,O,A).

We survey many interesting constructions on O-algebras that have this form. These include an augmentation ideal filtration of an augmented O-algebra A, the topological Andre-Quillen homology of A, the topological Hochschild homology of A, and the tensor product of A with a space. Right O-modules come with canonical increasing filtrations, and this leads to filtrations of all of the above.

In particular, I can show that a filtration on TAQ(A) defined recently by Behrens and Rezk agrees with one I defined about a decade ago, as was suspected. This is joint work with Luis Pereira.