# TTT119

## University of Warwick

## Date 25th April 2024

## Speakers

Johnny Nicholson (Glasgow)

Yuqing Shi (MPI)

Markus Szymik (Sheffield)

## Programme

All talks will be in B3.03, Zeeman Building, University of Warwick.

[Meet in the Common Room for those around before the first talk.]

12.00-13.00: Yuqing Shi

13.00-14.30: Lunch

14.30-15.30: Johnny Nicholson

15.30-16.00: Tea

16.00-16.30: Gong show (=short contributed talks: see below)

16.30-17.30: Markus Szymik

Early dinner at a restaurant near Coventry station.

Gong show: This is an opportunity to open some conversations. Please contact John Greenlees (John.Greenlees@warwick.ac.uk) if you would like to contribute. Format: (a) 8 minute slots, (b) blackboard only, (c) PhD students have priority, (d) selection at lunchtime on the day (signup possible until then).

## Titles and abstracts

Yuqing Shi (MPI, Bonn)

Title: Operadic Koszul duality and the costabilisation of O-algebras in the telescope localisation of spectra

Abstract: The theory of operadic Koszul duality can be thought of as the assignment of a cooperad K(O) to an operad O and vice versa, originally due to Ginzburg-Kapranov. This assignment at the level of operads and cooperads induces a functor CE_{O} from the category of O-algebras to the category of K(O)-coalgebras (sometimes called the Koszul duality functor). For example, let O be the Lie operad, this functor assigns a differential graded Lie algebra to its Chevalley-Eilenberg cochain complex. In this talk I will first recall the operadic Koszul duality in the language of (∞, 1)-categories. Then I will explain that the functor CE_{O} induces a comparison functor between the ∞-category of O-algebras and an inverse limit obtained from the ∞-category of K(O)-coalgebras. As an application, we obtain a result on the costabilisation, i.e. the stabilisation of the opposite ∞-category, of the ∞-category of O-algebras in the monochromatic layer of spectra. In the case where O is the spectral Lie operad, we obtain as a corollary the universal property of the Bousfield-Kuhn functor.

John Nicholson (Glasgow)

Title: Simple homotopy types of even dimensional manifolds

Abstract: Two CW-complexes are said to be simple homotopy equivalent if they are related by a sequence of collapses and expansions of cells. This notion interpolates between homeomorphism and homotopy in the sense that simple homotopy equivalent implies homotopy equivalent, and homeomorphic implies simple homotopy equivalent. It consequently proved extremely useful in manifold topology and is behind the s-cobordism theorem which is the basis for the vast majority of manifold classification results in dimension at least 4.

The aim of this talk will be to present the first examples of two 4-manifolds which are homotopy equivalent but not simple homotopy equivalent, as well as in all higher even dimensions. The examples are constructed using surgery theory and the s-cobordism theorem, and are distinguished using methods from algebraic number theory and algebraic K-theory. This is joint work with Csaba Nagy and Mark Powell. I will also discuss progress on the question of whether smooth 4-manifolds exist with these properties, through joint work with Daniel Kasprowski and Simona Veselá.

The talk will be accessible for a wide audience of topologists. In particular, I will assume no prior knowledge of surgery theory or simple homotopy equivalence.

Markus Szymik (Sheffield)

Title: A Burnside ring for racks

Abstract: Burnside rings for groups are an indispensable bookkeeping device when studying group actions. Various refinements of the idea allow us to adapt it to similar situations. In this talk, I will present an extension to the setting of racks and quandles, which can be thought of as actions without groups. It features joint work with Nadia Mazza.

## Further information

Participants from the UK nodes can claim travel expenses. There are links to the form on the main TTT page.

Many thanks to John Greenlees for organising TTT119.