Algebraic Topology Seminar

Clover May: Classifying modules of equivariant Eilenberg--MacLane spectra

Classically, since \( \mathbb{Z}/p \) is a field, any module over the Eilenberg—MacLane spectrum \( H\mathbb{Z}/p \) splits as a wedge of suspensions of \( H\mathbb{Z}/p \) itself. Equivariantly, cohomology and the module theory of \( G \)-equivariant Eilenberg--MacLane spectra are much more complicated.

For any \( p \)-group and the constant Mackey functor \( \mathbb{Z}/p \), there are infinitely many indecomposable \( H\mathbb{Z}/p \)-modules. Previous work together with Dugger and Hazel classified all indecomposable \( H\mathbb{Z}/2 \)-modules for the group \( G = C_2 \). The isomorphism classes of indecomposables fit into just three families. By contrast, joint work with Jacob Grevstad shows for \( G = C_p \) with \( p \) an odd prime, the classification of indecomposable \( H\mathbb{Z}/p \)-modules is wild.

Christian Carrick: Slice spectral sequences through synthetic spectra

We define a \( t \)-structure on the category of filtered \( G \)-spectra such that for a Borel \( G \)-spectrum \( X \), the slice filtration of \( X \) is the connective cover of the homotopy fixed-point filtration of \( X \). Using this, we show that the slice spectral sequence for the norm \( N_{C_2}^G MU_{\mathbb{\mathbb{R}}} \) of Real bordism theory refines canonically to a \( \mathbb{E}_\infty \)-algebra in \( MU \)-synthetic spectra, when \( G \) is a cyclic \( 2 \)-group. Concretely, this gives a map of multiplicative spectral sequences from the classical Adams--Novikov spectral sequence of \( \mathbb{S} \) to the slice spectral sequence for \( N_{C_2}^G MU_{\mathbb{\mathbb{R}}} \) that respects the higher \( \mathbb{E}_\infty \) structure, such as Toda brackets and power operations. We speculate further on a relationship to the equivariant ANSS based at Tom Dieck's homotopical complex bordism \( MU_G \).

Resources: [Video: YouTube] · [Video: Bilibili]

Jack Davies: Detection methods with synthetic spectra

The use of synthetic (or motivic or filtered or...) spectra for studying spectral sequences from a homotopical perspective has been a major theme in many recent computations in the stable homotopy groups of spheres. In this talk, I would like to discuss some basic tools and techniques for using synthetic spectra, paired with a detection spectrum \( X \), like real K-theory or topological modular forms, together with some operations on \( X \), to produce infinite periodic families in the stable homotopy groups of spheres. In particular, we want to focus on the utility of the synthetic Hurewicz image of \( X \) to detect these families in the classical stable homotopy groups of spheres. We will begin with the height one example of real K-theory together with its natural Adams operations in some detail, before moving onto various generalisations at height \( 2 \). If there is time, we will comment on some more recent advances. This is all joint work with Christian Carrick.

Resources: [Video: YouTube] · [Video: Bilibili] · [Seminar Notes]

Lucas Mann: The integral chromatic splitting conjecture via \(p\)-adic geometry

Special Location: This talk will be held in-person at 镜春园77201.

In 2024 Barthel--Schlank--Stapleton--Weinstein proved the rational version of the chromatic splitting conjecture, a deep conjecture on homotopy groups of spheres that previously seemed out of reach. Their method crucially uses \( p \)-adic geometry, specifically the isomorphism of the Drinfeld and Lubin--Tate towers. In a recent joint project initiated at an AIM workshop, we attack the integral (i.e. mod \( p \)) version of the splitting conjecture by similar methods. We construct a mod \( p \) version of the Lubin--Tate tower and Drinfeld tower and use them to relate the splitting conjecture to the computation of a certain \( p \)-adic cohomology on Drinfeld's upper half space. We then use the recent 6-functor formalism of Anschütz--Le-Bras--Mann to compute this cohomology. Our computation in particular predicts the existence of an unexpected error term in the splitting conjecture for small primes.

Kaif Hilman: Equivariant localizing motives for finite groups

In this talk, I will give a proposal for a definition of genuine equivariant localising motives for finite groups. This notion will be based on that of idempotent complete equivariantly stable categories. Using isotropy separation arguments on equivariant cubes and the recent insights of Ramzi-Sosnilo-Winges, we will see how to use this version of motives to enhance the algebraic K-theory functor with the structure of multiplicative norms. Time permitting, we will also discuss other applications such as showing that all genuine \(G\)-spectra are the K-theory of a \(G \)-stable category. This reports on joint work-in-progress with Maxime Ramzi.

Resources: [Video: YouTube] · [Video: Bilibili]

Ningchuan Zhang: Profinite transfers in \( K(n) \)-local homotopy theory

Special Location: This talk will be held in-person at 镜春园77201.

After \( K(1) \)-localization, the classical \( J \)-homomorphism in algebraic topology can be interpreted as a profinite transfer map. More precisely, it is a transfer map \( \Sigma^{-1}KO^\wedge_2 \to \mathbb{S}_{K(1)} \) from the \( C_2 \)-homotopy fixed points (with a twist) to the \( \mathbb{Z}_2^\times \)-homotopy fixed points of the \( 2 \)-complete complex topological \( K \)-theory. In joint work in progress with Guchuan Li, we extend this idea to define and study profinite transfers between homotopy fixed points of the Morava \( E \)-theory by closed subgroups of the Morava stabilizer group. We introduce two definitions of the profinite transfer maps. The first defines them as duals to the profinite restriction maps in the appropriate category. At large primes, we show that the image of the transfer map \( \Sigma^{-n^2}E_n \to \mathbb{S}_{K(n)} \) on homotopy groups is the \( n^2 \)-th filtration in the homotopy fixed point spectral sequence. A second definition is based on the 6-functor formalism for smooth representations of \( p \)-adic Lie groups after Heyer--Mann. We prove that the two definitions of profinite transfers are equivalent for homotopy fixed points of the Morava \( E \)-theory.

Andrew Baker: Endotrivial modules, Picard groups and chromatic Jokers

Associated to a finite dimensional cocommutative (graded) Hopf algebra there is a symmetric monoidal stable module category and a Picard group whose elements are endotrivial modules. These objects have been the focus of a lot of activity in  the following cases: finite group algebras over fields of positive characteristic, finite subHopf algebras of the Steenrod algebra \( \mathcal{A} \).  I will discuss these and then explain what the algebraic Joker module over \( \mathcal{A}(1) \) is and how it generalises to modules over the  \( \mathcal{A}(n) \) subHopf algebras. For small values of  \( n \) these can be realised as cohomology of spectra. Then I will explain how double Joker spectra when viewed in chromatic level 2 give rise to interesting endotrivial modules over \( \mathbb{F}_4Q_4 \), the group ring of the quaternion group of order 8.