# Warwick algebraic topology seminar 20/21

The current seminar is here.

###### Term 3

The talks will take place on Tuesday on MS teams at 4pm. You can join the team with the code hud33su, or email me at emanuele.dotto@warwick.ac.uk.

Date Speaker Affiliation Title Notes
Apr 27 Jonas McCandless Münster Cyclotomic spectra with Frobenius lifts and TR Video
May 4 Gabriel Angelini-Knoll Freie Universität Berlin Real Hochschild homology, norms, and Witt vectors Video
May 11 Markus Land Copenhagen On the K-theory of pullbacks and pushouts Video
May 18 Andrea Bianchi Copenhagen Parametrised moduli spaces of surfaces as infinite loop spaces Video
May 25 Nick Georgakopoulos Chicago The RO(G) graded cohomology of G-equivariant classifying spaces Video
Jun 1 Alexander Kupers Toronto Scarborough The Disc-structure space of a manifold Video
Jun 8 Mura Yakerson ETHZ Hermitian K-theory via oriented Gorenstein algebras Video
Jun 15 Alexander Berglund Stockholm Characteristic classes of manifold bundles and graph homology
Jun 22 Andrew Blumberg Columbia Waldhausen's algebraic K-theory chromatic convergence conjecture Video
Jun 29 Jack Davies Utrecht Adams operations on topological modular forms Video
##### Abstracts
###### Jonas McCandless: Cyclotomic spectra with Frobenius lifts and TR
​The cyclotomic trace is a map from algebraic K-theory to the more computationally accessible invariant given by topological cyclic homology, and in many cases this map is an equivalence after a suitable completion. The classical construction of topological cyclic homology relies on another invariant called TR which in turn is built from topological Hochschild homology using rather intricate methods from genuine equivariant homotopy theory. Recently, Nikolaus and Scholze gave a construction of topological cyclic homology completely bypassing the use of TR and genuine equivariant homotopy theory which simplified the theory to a great extent. Nonetheless, the importance of TR has persisted; for instance in the work of Antieau and Nikolaus on cyclotomic homotopy groups and in the work of Mathew on the Segal conjecture for topological Hochschild homology.
​In this talk, I will present a construction of TR which does not rely on genuine equivariant homotopy theory similar in spirit to the construction of topological cyclic homology given by Nikolaus and Scholze. In fact, the construction exhibits TR as the co-free cyclotomic spectrum with Frobenius lifts extending work of Krause and Nikolaus in the p-typical case. As an application, we give a formula for TR in terms of Bloch's spectrum of curves on K-theory generalizing work of Hesselholt, Betley, and Schlichtkrull.
###### Gabriel Angelini-Knoll: Real Hochschild homology, norms, and Witt vectors

By work of Angeltveit et al., topological Hochschild homology can be viewed as the norm from the trivial group to the circle group. Angeltveit et al. also give new definitions of norms from the n-th roots of unity to the circle and recently Blumberg-Gerhardt-Hill-Lawson used this to define new notions of Hochschild homology and Witt vectors for associative Green functors. In joint work with Teena Gerhardt and Mike Hill, we offer the perspective that Real topological Hochschild homology is the norm from the cyclic group of order two to the orthogonal group O(2). We then prove a multiplicative double coset formula for the norm extending a formula for real topological Hochschild homology by Dotto-Moi-Patchkoria-Reeh to dihedral groups. In my talk, I will discuss this perspective and how it motivates new definitions Real Hochschild homology and Witt vectors for algebras over the equivariant little disk operad in Mackey functors, which can be accessed using techniques in equivariant algebra.

###### Markus Land: On the K-theory of pullbacks and pushouts

I will report on joint work with Tamme. First, I will give a brief review of (a slight generalisation) of our previous theorem on the K-theory of pullbacks and will discuss some of its applications such as Suslin’s famous excision theorem. I will then indicate how, under certain assumptions, we can obtain a formula for the circle-dot ring appearing in our previous theorem as a pushout of E_1-ring spectra, and explain some examples to illustrate that the resulting pushouts can often be calculated explicitly, allowing for new calculations of K-theories to be performed. Using this new formula, I will give a new perspective on Waldhausen’s results about the K-theory of „generalised free products“ (which are simply E_1 pushouts) and indicate some consequences for the K-theory of certain coconnective ring spectra.

###### Andrea Bianchi: Parametrised moduli spaces of surfaces as infinite loop spaces

We consider the E_2-algebra ΛM_{∗,1} := \coprod_{g>=0} ΛM_{g,1} consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion ΩBΛM_{∗,1}: it is the product of Ω^∞ MTSO(2) with the free Ω^∞-space over a certain space X. This extends the classical result ΩBM_{∗,1}=Ω^∞ MTSO(2), due to Madsen and Weiss, to the setting of surface bundles parametrised over S^1.

I will define the space X in the statement and give a brief sketch of the proof, which combines two inputs: -on the one hand, we obtain a structure result for centralisers of mapping classes in generic mapping class groups Gamma_{g,n}, for g>=0 and n>=1: this uses standard techniques of the theory of mapping class groups, such as arc complexes; -on the other hand, we generalise the theory of operads with homological stability, developed recently by Tillmann et al., to the setting of coloured operads, and compute the group completion of certain "relatively free" algebras over such operads (with respect to a suboperad given by a family of groups); the main application involves a coloured version of Tillmann's surface operad.

This is joint work with Florian Kranhold and Jens Reinhold.

###### Nick Georgakopoulos: The RO(G) graded cohomology of G-equivariant classifying spaces

The cohomology of classifying spaces is an important classical topic in algebraic topology. However, much less is known in the equivariant setting, where one wants to know the RO(G)-graded cohomology of classifying G-spaces. The problem is that RO(G)-graded cohomology is notoriously difficult to compute even when G is cyclic. In this talk, I will explain my computations in the case of cyclic 2-groups G while keeping technical details to a minimum. The main goal is to understand rational equivariant characteristic classes, but I will also discuss some mod 2 computations and their relevance to the equivariant dual Steenrod algebra.

###### Alexander Kupers: The Disc-structure space of a manifold

Surgery theory attempts to understand the category of manifolds by mapping it to the category of topological spaces, with the goal of understanding the target and fiber of this map using homotopy-theoretic methods. In this talk I'll propose an alternative inspired by embedding calculus, mapping the category of manifolds to the category of simplicial presheaves on the category of discs, and explore its surprising features. This is joint work with Manuel Krannich.

###### Mura Yakerson: Hermitian K-theory via oriented Gorenstein algebras

In this talk, we will discuss a new geometric model for hermitian K-theory as a motivic space, given by the Hilbert scheme of Gorenstein subschemes of infinite affine space that are equipped with an orientation. This result gives a new universality property of hermitian K-theory as a generalized cohomology theory on smooth schemes. This is joint work with Marc Hoyois, Joachim Jelisiejew and Denis Nardin.

###### Alexander Berglund: Characteristic classes of manifold bundles and graph homology

We introduce new families of rational characteristic classes of fiber bundles with fiber a simply connected closed manifold. The new classes are associated to homology classes in a certain graph complex. The graph complex we consider is closely related to the Feynman transform of the Lie operad, the homology of which is known to be expressible in terms of the homology of automorphism groups of free groups.

The new classes may be viewed as a generalization of the Miller-Morita-Mumford classes; these are recovered from degree zero graph homology. When specialized to block bundles with fiber #^g S^d x S^d (the g-fold connected sum of S^d x S^d), we recover certain classes whose existence was implied by the calculation, due to Ib Madsen and myself, of the stable rational cohomology of the block diffeomorphism group (relative to a disk) of #^g S^d x S^d (2d>4). In particular, this implies that the family of characteristic classes associated to a non-trivial graph homology class is non-trivial

Our approach is based on rational homotopy theory and entails the construction of new rational models for classifying spaces of fibrations.

###### Andrew Blumberg: Waldhausen's algebraic K-theory chromatic convergence conjecture

In analogy with the chromatic convergence theorem of Hopkins and Ravenel, Waldhausen conjectured that the natural map K(S) \to \holim_n K(L_n S) is a weak equivalence. In this talk, I will describe joint work with Mandell and Yuan on this question. Along the way we give an explicit calculation of holim_n L_n K(S), the chromatic completion of K(S).

###### Jack Davies: Adams operations on topological modular forms

The classical construction of Adams operations on complex K-theory (KU) use the fact that there is a geometric interpretation of the cohomology classes of this cohomology theory using vector bundles. One does not have such a geometric description for elliptic cohomology theories, and this includes the universal‘‘ such theory Tmf, called topological modular forms. In this talk, we will discuss an alternative description of the stable Adams operations on KU in terms of some algebro-geometric input, and then use this as inspiration to define Adams operations on Tmf. We will see these operations have many of the expected and desired properties, and also discuss fundamental calculations and some further curiosities.

###### Term 2
Date Speaker Affiliation Title Notes
Jan 12 Maximilien Péroux University of Pennsylvania Coalgebras and comodules in stable homotopy theory Video
Jan 19 Herman Rohrbach Wuppertal When life gives you hyperbolic forms... Video
Jan 26 John Berman UT Austin Enriched K-theory and THH Video
Feb 2 Dan Berwick-Evans Urbana-Champaign How do field theories detect the torsion in topological modular forms? Video
Feb 9 Lior Yanovski Bonn Higher Cyclotomic Extensions in Chromatic Homotopy Theory Video
Feb 16 Clover May UCLA The unique commutative ring structure on rational equivariant K-theory Video
Feb 23 Severin Bunk Hamburg Universal Symmetries of Gerbes and a Smooth Model for the String Group Video
Mar 2 David Ayala Montana State Picard group of genuine-C_{p^n} spectra via Stratifications Video
Mar 9 Michael Ching Amherst Tangent ∞-categories and Goodwillie calculus Video
Mar 16 Tomer Schlank Jerusalem Galois extension in the Telescopic Category
##### Abstracts
###### Maximilien Péroux: Coalgebras and comodules in stable homotopy theory

In higher algebra, we study algebraic objects endowed with a multiplication that is associative only up to (coherent) homotopy, or commutative up to (coherent) homotopy. In this Brave new algebra, we study algebras and modules that includes the classical theory of algebra. The ground ring is not the ring of integer anymore, it is the sphere spectrum. Rigidification results (or sometimes called rectification) state that some of these highly coherent algebras over some rings can have their multiplication rigidified into a strictly associative multiplication. This has been used in many instances using the tool of model categories. In fact, in the 90s, many symmetric monoidal model categories of spectra were introduced such that strictly associative associative algebras were representing A_\infty-algebras, and similarly E_\infty-algebras.

In this talk, we will explore the dual algebraic objects of coalgebras and comodules in higher algebra. Instead of a multiplication, we have a comultiplication\coaction that we now require to be co-associative up to higher homotopy. I will show that higher algebras are enriched over higher coalgebras and thus, coalgebras provide insight on the structure for algebras. However, we will see that these objects are much more mysterious than algebras. I will show that none of the current monoidal model categories of spectra represent well the higher coalgebras in spectra. This is will hint that the correct language to study higher coalgebra is infinity-categories. I will also show that it is challenging but possible to rigidify coaction of comodules when using connective spectra over a field. This result allows to define a derived cotensor product of comodules which has not been possible before.

###### Herman Rohrbach: When life gives you hyperbolic forms...

Atiyah and Segal proved their famous completion theorem for topological K-theory back in 1969. In particular, it states that the natural map from the G-equivariant K-theory of the point (where G is a compact Lie group) to the K-theory of the classifying space BG is a completion of rings. Almost fifty years later, Krishna (2015) proved an analogue for algebraic K-theory. Whether and under what conditions Atiyah-Segal completion holds for Hermitian K-theory (Grothendieck-Witt theory) remains an open question, and is the topic of my PhD thesis. In this talk, I will sketch a proof of Atiyah-Segal completion for GW-theory in the simplest case of the geometric classifying space of a split torus, in a jungle of hyperbolic forms, long exact sequences and pro-modules...

###### John Berman: Enriched K-theory and THH

I will discuss applications (both established and conjectural) of enriched categories to topological Hochschild homology and K-theory. This builds on some new developments reducing the theory of enriched infinity categories to higher algebra.

###### Dan Berwick-Evans: How do field theories detect the torsion in topological modular forms?

Since the mid 1980s, there have been hints of a connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) for which cocycles are 2-dimensional (supersymmetric) field theories. Properties of these field theories lead naturally to the expected integrality and modularity properties of classes in TMF. However, the abundant torsion in TMF has always been mysterious from the field theory point of view. In this talk, we will describe a map from 2-dimensional field theories to a cohomology theory that approximates TMF. This map affords a cocycle description of certain torsion classes. In particular, we will explain how a choice of anomaly cancelation for the supersymmetric sigma model with target S^3 determines a cocycle representative of the generator of \pi_3(TMF)=Z/24.

###### Lior Yanovski: Higher Cyclotomic Extensions in Chromatic Homotopy Theory

In stable homotopy theory, the Morava K-theories K(n) play the role of "prime fields", which interpolate between characteristic 0 and characteristic p. The main approach for studying the K(n)-local categories is via their Galois extensions. These Galois extensions are governed by the Morava stabilizer groups, which are of an arithmetic nature. In a joint work with T. Schlank and S. Carmeli, we use higher semiadditivity (a.k.a ambidexterity) to reinterpret the abelian Galois extensions of the K(n)-local categories as "higher analogues" of cyclotomic extensions. We apply this perspective to lift these abelian Galois extensions to the far less understood T(n)-local categories and draw some conclusions regarding the Picard groups of these categories.

###### Clover May: The unique commutative ring structure on rational equivariant K-theory

The uniqueness of complex K-theory as a commutative ring spectrum was shown by Baker--Richter in 2005 using obstruction theory. Equivariantly, there are many possible levels of commutativity and these are encoded by norms. Using rational algebraic models, we show the uniqueness of rational equivariant KU as both a naive and genuine commutative ring spectrum for G a finite abelian group. We first calculate the image of KU as a naive commutative spectrum in the algebraic model of Barnes--Greenlees--Kedziorek, which is given by rational CDGAs with an action of the Weyl group. Then we extend this calculation to Wimmer's infinity-categorical model for genuine commutative spectra, which incorporates shadows of the norms. It is difficult to calculate the image of a spectrum in either model. However, in both cases, formality completes the computation and proves uniqueness. This is joint work with Anna Marie Bohmann, Christy Hazel, Jocelyne Ishak, and Magdalena Kedziorek.

###### Severin Bunk: Universal Symmetries of Gerbes and a Smooth Model for the String Group

Gerbes are geometric objects describing the third integer cohomology of a manifold and the B-field in string theory; they can essentially be understood as bundles of categories whose fibre is equivalent to the category of vector spaces. I will start by explaining gerbes and their categorical features. The main topic of this talk will then be the study of symmetries of gerbes from a universal perspective. We will see that these symmetries are completely encoded in a certain extension of smooth 2-groups. In the last part of the talk, I will survey how this construction can be used to provide a new smooth model for the string group, via a theory of principal bundles and group extensions in ∞-topoi.

###### David Ayala: Picard group of genuine-C_{p^n} spectra via Stratifications.

This talk will be centered around a recent computation, joint with Nick Rozenblyum and Aaron Mazel-Gee, of the Picard group of genuine-C_{p^n} spectra, for p an odd prime.

It will go something like this. Let G be a finite group, and let R be a commutative ring (or ring spectrum). We’ll examine genuine-G R-modules’’. One source of examples of such arise from the reduced chains C(S^V;R) with coefficients in R of the one-point compactification of a representation V of G. Such examples are hardly general, for they are tensor-invertible, or 1-dimensional’’. The Picard group of genuine-G R-modules consists of all such 1-dimensional objects. These naturally grade the homotopy groups of general genuine-G R-modules, sufficiently robustly to detect equivalences (think: Whitehead’s theorem). So we seek to identify this Picard group.

In the aforementioned joint work, based on early work of Greenlees-May, Glasman, and others, we articulate a sense in which the category of genuine-G R-modules shares many features with quasi-coherent sheaves over a stratified scheme, actually a stratified algebraic stack with very few points. I will supply a definition of a stratified non-commutative stack’’, and outline how both quasi-coherent sheaves on a stratified scheme as well as genuine-G R-modules organize as such an entity.

An advantage of this perspective is that it lends to certain computations by doing so stratum-wise, then assembling these results via gluing data among strata. I will demonstrate this computational approach by outlining the computation of the Picard group of genuine-C_{p^n} Z-modules.

###### Michael Ching: Tangent ∞-categories and Goodwillie calculus

This talk is about the framework of tangent categories, introduced by Rosický and Cockett-Cruttwell to axiomatize the tangent bundle functor on smooth manifolds. In joint work with Kristine Bauer and Matthew Burke, we have extended that framework to ∞-categories and constructed an important new example based on a tangent bundle construction of Lurie. We show that this example encodes Goodwillie's theory of functor calculus, making precise the analogy between Goodwillie calculus and the differential calculus of smooth manifolds. If there's time I will also describe a pair of tangent structures on ∞-toposes, one a restriction of the Goodwillie tangent structure and one an extension.

###### Tomer Schlank: Galois extension in the Telescopic Category

We shall employ the existence of so-called higher cyclotomic Galois extensions in the T(n)-local category to get several results about it. In particular, we shall discuss Picard elements, non-abelian Galois extension, Higher Kummer theory, and Fourier transform. If time permits we shall also discuss a so-called "Cyclotomic Redshift " in algebraic K-theory, These are joint projects with Lior Yanovski, Shachar Carmeli, Tobias Barthel, and Shay ben Moshe.

###### Term 1

The talks will take place on Tuesday on MS teams at 4pm. You can join the team with the code hud33su, or email me at emanuele.dotto@warwick.ac.uk.

Date Speaker Affiliation Title Notes
Oct 6 Lennart Meier Utrecht Groups and Group Laws Behind Equivariant Cohomology Theories Video
Oct 13 Dylan Wilson Harvard The multiplication on truncated Brown-Peterson spectra Video
Oct 20 Stefano Ariotta Münster Filtered objects and coherent chain complexes in stable ∞-categories Video
Oct 27 Alice Hedenlund Oslo Multiplicative spectral sequences via decalage Video
Nov 3 Ieke Moerdijk Utrecht Homology of infinity-operads Video
Nov 10 Irakli Patchkoria Aberdeen On the Balmer spectrum of derived Mackey functors Video
Nov 17 Danica Kosanović Paris XIII Knot invariants from homotopy theory Video
Nov 24 Asaf Horev Stockholm Genuine equivariant factorization homology Video
Dec 1 Goncalo Tabuada Warwick Motivic Atiyah-Segal completion theorem Video
Dec 8 Elden Elmanto Harvard

On Bass' NK groups in mixed and positive characteristics

Video
##### Abstracts
###### Lennart Meier: Groups and Group Laws Behind Equivariant Cohomology Theories

Formal groups play a central role in non-equivariant chromatic homotopy theory. We explore analogues in the global equivariant world, starting our tour with equivariant K-theory and touching in particular upon equivariant elliptic cohomology and the new concept of spectral global group laws. This is based on joint work with David Gepner and with Markus Hausmann.

###### Dylan Wilson: The multiplication on truncated Brown-Peterson spectra

(Joint with Jeremy Hahn). The truncated Brown-Peterson spectra, BP<n>, are important objects in chromatic homotopy theory. They are known to be associative and, by theorems of Lawson and Senger, not to be commutative, in general. We produce, for each n, a form of BP<n> which is an E_3-algebra over complex cobordism. The construction uses higher centralizers, Koszul duality, and a curious highly structured algebra over complex cobordism. Our result, combined with work of Bruner-Rognes and unpublished work of Krause-Nikolaus, has implications for the topological cyclic homology and K-theory of truncated Brown-Peterson spectra.

###### Stefano Ariotta: Filtered objects and coherent chain complexes in stable ∞-categories

Given any stable ∞-category equipped with a t-structure, a very general construction, whose first incarnation is due to Beilinson, lets one endow its
∞-category of filtered objects with an induced t-structure. In current work in progress, we give a new perspective on such t-structures of Beilinson type by
showing that (complete) filtered objects are equivalent to Joyal's "coherent chain complexes" and by studying the formal properties enjoyed by this equivalence.

###### Alice Hedenlund: Multiplicative spectral sequences via decalage

Décalage was originally introduced by Deligne and one can roughly describe the process as providing us with a way to encode “turning the page of a spectral sequence” on the level of filtrations. Although not originally phrased in this way, décalage can be made sense in terms taking connective covers of a filtration in a certain t-structure on the category of filtered complexes called the Beilinson t-structure. This allows one to generalise the construction also to filtered objects in other stable oo-categories, such as spectra. In this talk, we show that the language of the Beilinson t-structure and décalage provides access to highly structured results on filtered spectra and their associated spectral sequences. In particular, we sketch how it can be used to show that the functor assigning a spectral sequence to a filtered spectrum can be endowed with the structure of a map of oo-operads.

###### Ieke Moerdijk: Homology of Infinity-Operads

We will describe a natural and elementary way of extending the homology of operads to the case of operads-up-to-homotopy, alias oo-operads. We will also discuss various ways to extend the theory: (a) The domain of definition can be extended to obtain a left Quillen functor on dendroidal complete Segal spaces. (b) The values can be enriched to have the structure of an oo-cooperad. The latter construction has an inverse up to homotopy, from oo-cooperads to oo-operads, which extends to bar-cobar duality of Ginzburg-Kapranov and Getzler-Jones.

The talk is based on joint work with Eric Hoffbeck.

###### Irakli Patchkoria: On the Balmer spectrum of derived Mackey functors

A result of Devinatz-Hopkins-Smith describes the spectrum of prime ideals of finite spectra. Under this identification the information encoded by the zeroth and infinite chromatic levels can be identified with the spectrum of the derived category of integers, which by a result of Hopkins-Neeman is just equivalent to Spec(Z). It turns out that in the equivariant context neither the spectrum of the usual derived category of Mackey functors nor the spectrum of the Burnside ring play the role of Spec(Z). Given a finite group G, we show that Kaledin's category of derived G-Mackey functors describes the zeroth and infinite chromatic levels of the Balmer spectrum of finite G-spectra. We compute the Balmer spectrum of derived G-Mackey functors. Along the way we will identify Kaledin's category with the homotopy category of the stable infinity category of HZ-linear spectral Mackey functors in the sense of Barwick. This is all joint with B. Sanders and C. Wimmer.

###### Danica Kosanović: knot invariants from homotopy theory

Embedding calculus of Goodwillie and Weiss is a certain homotopy theoretic technique for studying spaces of embeddings. When applied to the space of knots this method gives a sequence of knot invariants which are conjectured to be universal Vassiliev invariants. This is remarkable since such invariants have been constructed only rationally so far and many questions about possible torsion remain open. In this talk I will present some explicit computations and outline why these knot invariants are surjections. This confirms one half of the universality conjecture, and confirms it rationally, and p-adically in a range. We also prove some missing cases of the Goodwillie--Klein connectivity estimates.

###### Hasaf Horev: Genuine equivariant factorization homology

Factorization homology is a natural invariant of manifolds, a variant of Salvatore’s configuration space with summable labels. In dimension one this geometric construction recovers topological Hochschild homology. I’ll describe an equivariant extension of this theory, where the manifold, the coefficient and the resulting invariant all admit a finite group action. After discussing equivariant nonabelian Poincaré duality I’ll explain how one can use its interaction with equivariant Thom spectra to compute the real topological Hochschild homology of HF_2. Joint work with Inbar Klang and Foling Zou.

###### Goncalo Tabuada: Motivic Atiyah-Segal completion theorem

The famous Atiyah-Segal completion theorem describes Borel’s K-theory as the completion of equivariant K-theory with respect to the augmentation ideal. This topological result, proved in the late sixties, was adapted to the algebraic setting by Thomason in the eighties and by Krishna in the last few years. In this talk I will describe a motivic Atiyah-Segal completion theorem which applies not only to K-theory but also to other invariants such as periodic cyclic homology. Among other applications, it leads to an improvement of the original results of Krishna, Thomason and Atiyah-Segal.

This talk is based on a recent joint work with Michel Van den Bergh (arXiv:2009.08448).

###### On Bass' NK groups in mixed and positive characteristics

In characteristic zero, work of Cotinas-Haesemeyer-Weibel-Walker has elucidated the structure of Bass' NK groups - the obstruction to K-theory being homotopy invariant. I will report some joint work in progress with Martin Speirs on these NK groups in mixed and positive characteristics. Notable results include: a bounded p-torsion result in positive characteristics and a purity result in mixed characteristics.