TTT123

Date Thursday 4th December 2025

Speakers

The speakers will be Jack Davidson (Sheffield), Nathan Lockwood (Warwick) and Lewis Stanton (Southampton).

Programme

10.15-11.00 (Common Room):   Coffee

11.00-12 noon (B3.01):                Nathan Lockwood

14.00-15.00 (B3.01):                     Lewis Stanton

15.00-16.00 (Common Room):   Coffee 

16.00-17.00 (C1.14):                     Jack Davidson

Titles and abstracts

Nathan Lockwood (Warwick)

Title: The Z/2 Geometric Fixed Points of Real Cyclotomic Spectra

Abstract: Antieau and Nikolaus introduced topological Cartier modules to find the heart of a t-structure on cyclotomic spectra. In particular, THH of a perfect field K lies in the heart as the Witt vectors over K. 

The aim of the talk is to explain how to extend this theory to real cyclotomic spectra. There is a real version of THH for ring spectra with anti-involution, which is canonically a real cyclotomic spectrum. Further, Real Topological Cyclic Homology (TCR) is representable in the category of real cyclotomic spectra by work of Quigley-Shah. Following the observation that the Z/2 geometric fixed points of TCR only depend on the Z/2 geometric fixed points of the real cyclotomic spectrum, we will define a category modelling such geometric fixed points and show that the geometric fixed-points of TCR are representable in this category.  We will define a canonical t-structure on real cyclotomic spectra and real analogues of topological Cartier modules to identify the heart of  this t-structure.

Lewis Stanton (Southampton)

Title: Anick's conjecture for polyhedral products

Abstract: Anick conjectured the following after localisation at any sufficiently large prime - the pointed loop space of any finite, simply connected CW complex is homotopy equivalent to a finite type product of spheres, loops on spheres, and a list of well-studied torsion spaces defined by Cohen, Moore and Neisendorfer. We study this question in the context of moment-angle complexes, a central object in toric topology which are indexed by simplicial complexes. These are a special case of a family of spaces known as polyhedral products, which unify constructions across mathematics. Recently, much work has been done to find families of simplicial complexes for which the corresponding moment-angle complex satisfies Anick's conjecture integrally. In this talk, I will survey what is known and show that the loop space of any moment-angle complex is homotopy equivalent to a product of looped spheres after localisation away from a finite set of primes. This is then used to show Anick's conjecture holds for a much wider family of polyhedral products. This talk is based on joint work with Fedor Vylegzhanin.

Speaker: Jack Davidson (Sheffield)

Title: Extensions of reflexive homology and operads

Abstract: An oriented group is a discrete group G with a homomorphism from $G$ to the cyclic group of order two. Koam and Pirashvili studied cohomology theories for oriented algebras, i.e. associative algebras equipped with an action of the oriented group G (where elements of G act via (anti)-automorphisms, depending on their image in the cyclic group of order two). I will describe an extension of their theory using the framework of functor (co)homology and crossed simplicial groups, viewing it as an extension of reflexive homology (which is a homology theory for algebras with an anti-involution). We will demonstrate some basic properties of these theories and show how our framework allows us to identify them as "homotopical objects", i.e. as the homology of algebras over an operad. As a special case, we can describe reflexive homology as operadic homology. If time permits I will discuss the relationship between our theory and the recent work on "topological crossed simplicial group homology" of Angelini-Knoll -- Merling - Péroux. This is all joint work with Dan Graves and Sarah Whitehouse.