# Warwick Algebraic Topology Seminar 22/23

A list of the seminar talks of the previous years can be found here.

###### Term 2

The talks will take place on Tuesday at 4pm. Some of the talks will be in person in B3.02, some will be given by a remote speaker on MS Teams and streamed in B3.02.

Date | Speaker | Affiliation | Title | Notes |
---|---|---|---|---|

Jan 10 | Ismael Sierra | University of Cambridge | Homological stability of diffeomorphism groups using Ek algebras | B3.02 |

Jan 17 | Foling Zou | University of Michigan | Nonabelian Poincare duality theorem in equivariant factorization homology | TeamsLink opens in a new window/B3.02 |

Jan 24 | No Seminar | |||

Jan 31 | Neil Strickland | University of Sheffield | Questions around chromatic splitting | B3.02 |

Feb 7 | Luca Pol | University of Regensburg | Quillen stratification in equivariant homotopy theory | Teams/B3.02 |

Feb 14 | No Seminar | |||

Feb 21 | Constanze Roitzheim | University of Kent | How algebraic is a stable model category? | B3.02 |

Feb 28 | Gonçalo Tabuada | University of Warwick | B3.02 | |

Mar 7 | Ben Briggs | University of Copenhagen | B3.02 | |

Mar 14 | Jeffrey Carlson | Imperial College | B3.02 |

##### Abstracts

###### Ismael Sierra: Homological stability of diffeomorphism groups using Ek algebras

I will state some recent results about homological stability of diffeomorphism groups of manifolds and give an outline of their proof. In particular, I will talk about the connection to Ek algebras, and about certain complexes, called "splitting complexes", whose high-connectivities are essential to the proof. Finally I will sketch the proof of the high-connectivity of the splitting complexes, which is the most substantial part of the whole argument.

###### Foling Zou: Nonabelian Poincare duality theorem in equivariant factorization homology

The factorization homology are invariants of $n$-dimensional manifolds with some fixed tangential structures that take coefficients in suitable $E_n$-algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group $G$ by monadic bar construction following Kupers--Miller. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we compute the equivariant factorization homology on equivariant spheres for certain Thom spectra. In particular, we recover a computation of the Real topological Hoschild homology by Dotto--Moi--Patchkoria--Reeh.

###### Neil Strickland: Questions around chromatic splitting

The Chromatic Splitting Conjecture of Hopkins states that if we take the sphere spectrum, localise with respect to the Morava K-theory K(n), then localise again with respect to E(n-1), then the result splits as a coproduct of 2^n pieces, each of which is a sphere localised with respect to E(m) for some m < n. This is known to be false (by work of Beaudry) when n=p=2, but remains open (and mysterious) in general. We will explain how the conjecture, when combined with known phenomena such as chromatic fracture squares, predicts some calculations with a very intricate combinatorial/algebraic structure. These calculations appear to be self-consistent, which could easily have failed to be the case; this suggests that the conjecture may be true, or closely related to the truth.

###### Luca Pol: Quillen stratification in equivariant homotopy theory

Quillen’s celebrated stratification theorem provides a geometric description of the Zariski spectrum of the cohomology ring of any finite group with coefficients in a field in terms of information coming from its elementary abelian p-subgroups. The goal of this talk is to discuss an extension of Quillen's result to the world of equivariant tensor-triangular geometry. For the category of equivariant modules over a commutative equivariant ring spectrum we obtain a stratification result in the terms of the geometric fixed points equipped with their Weyl-group actions for all subgroups, and hence a classification of localizing tensor ideals. Finally, I will apply these methods to several examples of interests such as Borel-equivariant Morava E-theory and equivariant topological K-theory. This is joint work with Tobias Barthel, Natalia Castellana, Drew Heard and Niko Naumann.

###### Constanze Roitzheim: How algebraic is a stable model category?

There are many different notions of "being algebraic" used in stable homotopy theory. The relationships between those turn out to be unexpectedly subtle. We will explain the different ways in which a model category of interest can be algebraic, explore the different implications between them and illustrate those with plenty of examples.

(This is joint work with Jocelyne Ishak and Jordan Williamson.)

###### Term 1

Date | Speaker | Affiliation | Title | Notes |
---|---|---|---|---|

Oct 11 | Sebastian Chenery | Southampton | On Pushout-Pullback Fibrations | B3.03 |

Oct 18 | Thomas Read | Warwick | G-typical Witt vectors with coefficients and the norm | B3.03 |

Oct 25 | Severin Bunk | Oxford | Functorial field theories from differential cocycles | B3.03 |

Nov 1 | No Seminar | |||

Nov 8 | Thibault Décoppet | Oxford | Fusion 2-Categories associated to 2-groups | B3.03 |

Nov 15 | Foling Zou | University of Michigan | Nonabelian Poincare duality theorem in equivariant factorization homology | TeamsLink opens in a new window/B3.03 |

Nov 22 | Irakli Patchkoria | Aberdeen | Morava K-theory of infinite groups and Euler characteristic | B3.03 |

Nov 29 | Florian Naef | Trinity College Dublin | Relative intersection product, Whitehead-torsion and string topology | B3.03 |

Dec 6 | Lucy Yang | Harvard | A real Hochschild--Kostant--Rosenberg theorem | TeamsLink opens in a new window/B3.03 |

##### Abstracts

###### Sebastian Chenery: On Pushout-Pullback Fibrations

We will discuss recent work inspired by a paper of Jeffrey and Selick, where they ask whether the pullback bundle over a connected sum can itself be homeomorphic to a connected sum. We provide a framework to tackle this question through classical homotopy theory, before pivoting to rational homotopy theory to give an answer after taking based loop spaces.

###### Thomas Read: G-typical Witt vectors with coefficients and the norm

The norm is an important construction on equivariant spectra, most famously playing a key role in the work of Hill, Hopkins and Ravenel on the Kervaire invariant one problem. Witt vectors are an algebraic construction first used in Galois theory in the 1930s, but later finding applications in stable equivariant homotopy theory. I will describe a new generalisation of Witt vectors that can be used to compute the zeroth equivariant stable homotopy groups of the norm $N_e^G Z$, for $G$ a finite group and $Z$ a connective spectrum.

###### Severin Bunk: Functorial field theories from differential cocycles

In this talk I will demonstrate how differential cocycles give rise to (bordism-type) functorial field theories (FFTs). I will discuss some background on smooth FFTs, differential cohomology and higher gerbes with connection as a geometric model for differential cocycles before explaining the general principle for how to obtain smooth FFTs from higher gerbes. In the second part, I will focus on the two-dimensional case. Here I will present a concrete, geometric construction of two-dimensional smooth FFTs on background manifolds, starting from gerbes with connection. This is related to WZW theories. If time permits, I will comment on an extension of this construction which produces open-closed field theories.

###### Thibault Décoppet: Fusion 2-Categories associated to 2-groups

Motivated by the cobordism hypothesis, which provides a correspondence between fully dualizable objects and fully extended framed TQFTs, it is natural to seek out interesting examples of fully dualizable objects. In dimension four, the fusion 2-categories associated to 2-groups are examples of fully dualizable objects. In my talk, I will begin by reviewing the 2-categorical notion of Cauchy completion, and recall the definition of a fusion 2-category in detail. Then, I will explain how one can construct a fusion 2-category of 2-vector spaces graded by 2-group, and how this construction can be twisted using a 4-cocycle. Finally, it is important to understand when two such fusion 2-categories yield equivalent TQFTs. The answer is provided by the notion of Morita equivalence between fusion 2-categories, which will be illustrated using some examples.

###### Foling Zou: Nonabelian Poincare duality theorem in equivariant factorization homology

The factorization homology are invariants of $n$-dimensional manifolds with some fixed tangential structures that take coefficients in suitable $E_n$-algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group $G$ by monadic bar construction following Kupers--Miller. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we compute the equivariant factorization homology on equivariant spheres for certain Thom spectra. In particular, we recover a computation of the Real topological Hoschild homology by Dotto--Moi--Patchkoria--Reeh.

###### Irakli Patchkoria: Morava K-theory of infinite groups and Euler characteristic

Given an infinite discrete group G with a finite model for the classifying space for proper actions, one can define the Euler characteristic of G and the orbifold Euler characteristic of G. In this talk we will discuss higher chromatic analogues of these invariants in the sense of stable homotopy theory. We will study the Morava K-theory of G and associated Euler characteristic, and give a character formula for the Lubin-Tate theory of G. This will generalise the results of Hopkins-Kuhn-Ravenel from finite to infinite groups and the K-theoretic results of Adem, Lück and Oliver from chromatic level one to higher chromatic levels. Along the way we will give explicit computations for amalgamated products of finite groups, right angled Coxeter groups and certain special linear groups. This is all joint with Wolfgang Lück and Stefan Schwede.

###### Florian Naef: Relative intersection product, Whitehead-torsion and string topology

Given a closed oriented manifold one can define an intersection product on the homology. This can be extended to local coefficient, and further made relative to the diagonal. I will explain how such a relative self-intersection product is not homotopy invariant (in contrast to the ordinary intersection product) and how this is picked up by string topology. Eventually, we will identify the error term with the trace of Whitehead torsion. More precisely, we will extract an invariant from a Poincare embedding of the diagonal (in the sense of J. Klein) that is the trace of (a version of) Reidemeister torsion. This is based on joint work with P. Safronov.

###### Lucy Yang: A real Hochschild--Kostant--Rosenberg theorem

Grothendieck--Witt and real K-theory are enhancements of K-theory in the presence of duality data. Similarly to ordinary K-theory, real K-theory admits homological approximations, known as real trace theories. In this talk, I will identify a filtration on real Hochschild homology and compute the associated graded in terms of an analogue of de Rham forms. We will see how C₂ genuine equivariant algebra is the natural setting for these theories, provide equivariant enhancements of the cotangent and de Rham complexes, and sketch the proof of the main theorem. This work is both inspired by and builds on that of Raksit.