# FREUDENTHAL TOPOLOGY SEMINAR

## (organized with Ryan Quinn)

The main goal of the Freudenthal seminar is to foster a good atmosphere, and to encourage discussion between the participants. As such, each speaker may choose to talk on any topic they wish. The hope is that this exposes us to a wide range of material, and allows each speaker to give a talk on a topic they are enthusiastic about.

To ensure the audience still benefits, the speaker should focus on distilling the key idea and telling a good story. The following kinds of talks are encouraged:

Giving a talk on a paper whose proof uses a technique that may be applied more widely. Focus on extracting the key idea from the technique.

Giving a talk on a newish paper to highlight an interesting new result and potential applications. Focus on the result, and giving a good example that captures the key point of the result.

Giving a talk on a newish paper that uses a new technique that may applied more widely.

Giving a talk with a bird's-eye view on an specific area.

In spring 2024, the seminar will take place every Tuesday from 12:00 to 13:00 in HFG 6.10.

If you would like to give a talk please email me!

# SPRING 2024

Yuan's model for spaces

A classical type of problem in homotopy theory is to build algebraic (i.e. stable) models for a given unstable category. The prototypical example of this problem is the following question: what kind of extra structure can we put on the suspension spectrum of a space in order to uniquely characterize its homotopy type?

In this talk I will explain how Yuan [4] models (finite, simply connected) spaces in terms of their (dual) suspension spectra equipped with a structure of Frobenius-fixed E∞-algebra, building this up from the works of Sullivan [3] and Mandell [2] and the ideas of Nikolaus [1].

[1] T. Nikolaus, Frobenius homomorphisms in higher algebra.

[2] M. Mandell, E∞-algebras and p-adic homotopy theory.

[3] D. Sullivan, Infinitesimal computations in topology.

[4] A. Yuan, Integral models for spaces via the higher Frobenius.

The homotopy exponent of the 3-sphere

The famous Cohen-Moore-Neisendorfer theorem [1] says that the homotopy groups of S^{2n+1} contain no p^j-torsion elements for j > n. Gray had already constructed [2] elements of precisely this order using the image of J, so this bound is sharp. Their proof is an induction, relying on the base case n = 1, that is, that the p-local homotopy groups of S^3 consist only of (the copy of Z in degree 3 that you all know and love and) a bunch of Z/p-summands. This result, due to Selick [3], is a couple of years older and structurally quite different (decompositions of looped Moore spaces are nowhere in sight).

I will say something about Selick’s theorem and its proof.

[1] F. R. Cohen, J. C. Moore and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres.

[2] B. Gray, Unstable families related to the image of J.

[3] P. S. Selick, Odd primary torsion in π_k(S^3).

Homotopy-coherent algebraic structures and free Segal objects

Francesca Pratali (Université Sorbonne Paris Nord)

Notes [and an expanded version]

In [1], Chu and Haugseng describe a general framework for defining homotopy-coherent algebraic structures as presheaves on certain infinity-categories (called ‘algebraic patterns’) satisfying a Segal-type limit condition. Among the instances of this construction are E_n-algebras, (∞,n)-categories, dendroidal ∞-operads and Lurie’s ∞-operads. A systematic study of the theory of algebraic patterns permits to define, for example, enrichment of these objects, as well as to provide conditions under which Segal objects are preserved by left and right Kan extensions. In particular, it is possible to write an explicit formula for free Segal objects on certain well-behaved algebraic patterns.

In this talk, I will give an overview of this general setting, providing a wide range of examples. I will then focus on some of the techniques used for the computation of the free Segal object functor.

[1] H. Chu and R. Haugseng, Homotopy-coherent algebra via Segal condition.

Equivariant bordism without manifolds

This talk is meant both as an advertisement for and a leisurely introduction to global stable homotopy theory. I will begin by giving a crash course on equivariant and global spectra, with a particular focus on the algebraic structure present on their homotopy groups. I will then demonstrate how one can use this structure to deduce non-trivial ring theoretic properties for the coefficient rings of U(n)-equivariant complex bordism in an almost purely formal way.

[1] S. Schwede, Global homotopy theory.

[2] S. Schwede, Splittings of global Mackey functors and regularity of equivariant Euler classes.

[3] S. Schwede, Chern classes in equivariant bordism.

Pro-étale homotopy types of schemes

Sebastian Wolf (Universität Regensburg)

In this talk, we will introduce a refinement of Artin-Mazur-Friedlander's étale homotopy type of a scheme, using the formalism of condensed mathematics. Furthermore, we will explain how this invariant is connected to the Galois-category of a scheme, introduced by Barwick-Glasman-Haine [1], and explore some consequences of this connection.

[1] C. Barwick, S. Glasman and P. Haine, Exodromy.

Segalification of simplicial spaces and some applications

Complete Segal spaces were first introduced in Rezk's seminal 1998 paper [4], which were later shown to model ∞-categories by Joyal-Tierney in [3]. A particular construction described by Rezk is an explicit completion functor which produces from a Segal space an equivalent complete Segal space. In recent work [1] by Barkan-Steinebrunner, they describe a formula for the Segal space freely generated by a simplicial space via the work of Dugger-Spivak [2] on simplicial necklaces. The aim of this talk is to explain this description of Segalification and present some consequences of this construction.

[1] S. Barkan and J. Steinebrunner, Segalification and the Boardmann-Vogt tensor product.

[2] D. Dugger and D. Spivak, Rigidification of quasi-categories.

[3] A. Joyal and M. Tierney, Quasi-categories vs Segal spaces.

[4] C. Rezk, A model for the homotopy theory of homotopy theories.

Class field theory for topologists

The goal of class field theory is to compute the abelianisation of the absolute Galois group of a global or local field. Since the 1950s, the proofs are carried out entirely in the language of Galois cohomology, consisting of a formal part ('abstract class field theory') and a few (long) computations. I will give an introduction to the area and explain the formal part of the story, using the formalism of Mackey functors from equivariant stable homotopy theory.

Cofiber tau philosophy and Hopf invariant one

In recent years, computations in the classical Adams spectral sequence have advanced tremendously via the use of deformations of the category of spectra, such as motivic spectra, equivariant spectra, and synthetic spectra. From the point of view of computations, these categories allow one to run various spectral sequences simultaneously, so that one may exploit the best features of each spectral sequence at the same time. The study of the Adams spectral sequence began with Adams' work on the Hopf invariant one problem, which he solved by producing a family of nontrivial d_2 differentials in the Adams spectral sequence. As a toy example to demonstrate these new tools, I will show how to deduce these d_2 differentials from calculations in BP-synthetic spectra.

Profinite descent for Picard groups

An important invariant of a symmetric monoidal ∞-category is its Picard group. The Picard group of K(n)-local spectra was first studied by Hopkins-Mahowald-Sadofsky, and despite the great efforts of many others since then, this group is still unknown at heights above 2. Recently, Mor [1] has described an approach to computing the K(n)-local Picard group using methods of descent. Descent is a well-established technique for computing Picard groups, but to apply it to the K(n)-local setting, one must carefully take its profinite nature into account. Mor does this using the language of condensed mathematics. In this talk, I will give an introduction to descent methods for Picard groups, and outline how condensed mathematics can be used to set up a spectral sequence converging to the K(n)-local Picard group.

[1] I. Mor, Picard and Brauer groups of K(n)-local spectra via profinite Galois descent.

The level n-elliptic genus

Ryan Quinn

In [1], Hirzebruch constructed the level n elliptic genus. In [2], Meier constructed connective modules of topological modular forms with level structures, and show that the Hirzebruch level n elliptic genus lifts to a map of ring spectra. Recently, in [3], Senger has shown that this further lifts to an E∞-ring map. In this talk, I will give a brief overview of [3].

[1] F. Hirzebruch, Elliptic genera of level N for complex manifolds.

[2] L. Meier, Connective models for topologlical modular forms of level n.

[3] A. Senger, Obstruction theory and the level n elliptic genus.

Easter break

Happy holidays everyone!

Cochain models for spaces

Let R be a commutative ring and X a space. There are many ways to endow the R-valued cochains C*(X;R) with extra structure (e.g. a cdga, or a cosimplicial R-algebra) in the hopes of turning C*(-;R) into a homotopically fully faithful functor on a suitable class of spaces (depending on R). I'll briefly review some classical work of Quillen, Sullivan, and Mandell and then try to get to the recent papers of Antieau [1] and Horel [2]. These work in the case R = Z and show that the cochains, interpreted as a "derived binomial ring", give a fully faithful description of nilpotent spaces of finite type.

[1] B. Antieau, Spherical Witt vectors and integral models for spaces.

[2] G. Horel, Binomial rings and homotopy theory.

Conference on (∞,n)-categories and their application

In the place of the seminar, you're very welcome to attend the all-week-long conference on (∞,n)-categories and their applications organised by Lennart Meier and Jaco Ruit.

The homotopy theory from Pursuing stacks

Despite its title, Grothendieck's very influential Pursuing Stacks is only partially concerned with higher stacks and out of 140 sections of the manuscript, more than 120 are dedicated to the foundations of homotopy theory. Starting with cohomological intuition coming from the theory of toposes, the point of view taken on homotopy theory is original and yet very enlightening.

In this talk, I will try to present some of these ideas and convey this (co)homological intuition. In particular, I will show how it is very natural to consider Cat as the "fundamental" model for homotopy types without any need for topological spaces or simplicial sets. Time permitting, I will also present the theory of derivators, which is a very convenient synthesis of homotopical and homological algebra

Atiyah's Real K-Theory

Atiyah's Real K-Theory is a C2-spectrum with fixed points KO and underlying spectrum KU. We present a modern-style construction of it and give streamlined proofs of its basic properties.

The coalgebraic enrichment of algebras

Maximilien Péroux (Michigan State University)

In the 60s, Sweedler showed that algebras form a category enriched in coalgebras, where the hom-set of algebra homomorphisms is replaced by a coalgebra of partial homomorphisms called measurings. This result has been extended in many contexts, notably in higher categories: I will show that ring spectra are enriched in coalgebra spectra. I will also present an application from joint work with Paige Randall North where we apply the coalgebraic enrichment as a way to gauge partial induction arguments for algebras over endofunctors.

UGC Gongshow

In the place of the seminar you're very welcome to attend the UGC Gongshow.

Comodules: the abelian, the derived, and the stable

In algebraic topology, comodules over a Hopf algebroid play an important role in the computation of homotopy groups, as they are the building blocks of the second page of Adams spectral sequences. Classically, to compute Ext groups of such comodules, various change-of-rings theorems have been proved.

This talk will be an introduction to comodules, focusing on the categorical perspective. I will discuss the results of Hovey and Strickland [1] that show that these change-of-rings theorems come from the structure of the abelian category of comodules. Next, we will discuss Hovey’s stable comodule category [2]. This can be thought of as an ∞-category that is a better alternative to the derived ∞-category of comodules. Some of the structure theorems from the abelian case upgrade to one of this stable version [3,4]. If time permits, I will briefly discuss the connection between the stable comodule category and synthetic spectra.

[1] M. Hovey and N. Strickland, Comodules and Landweber exact homology theories.

[2] M. Hovey, Homotopy theory of comodules over a Hopf algebroid.

[3] M. Hovey, Chromatic phenomena in the algebra of BP*BP-comodules.

[4] T. Barthel, D. Heard and G. Valenzuela, Local duality in algebra and topology.

Equivariant Unitary Bordism & Birational Invariants of Finite Groups

Bernardo Uribe (Universidad del Norte)

In this talk I will summarize what is known about torsion invariants on the Equivariant and Unitary Bordism groups, as well as the mysterious relation between the torsion invariants on surfaces and some birational invariants of finite groups.

Summer break

Happy holidays everyone!

# FALL 2023

The Alexander trick for homology spheres

Galatius and Randal-Williams have shown in a recent paper [1] that the group of homeomorphisms / diffeomorphisms fixing the boundary of a contractible d-dimensional manifold M (for d at least 6) has the same homotopy type of the group of homeomorphisms / diffeomorphisms fixing the boundary of the standard d-dimensional disk. As a consequence, this result implies that the group of homeomorphisms of M fixing the boundary is a contractible space, via the usual Alexander trick.

In this talk I will present one of the proofs in this paper, which makes use of the embedding calculus of Goodwillie and Weiss.

[1] S. Galatius and O. Randal-Williams, The Alexander trick for homology spheres.

Algebraicity and monoidal algebraicity of chromatic homotopy theory

For a chromatic height n and a prime p, consider the homotopy category of p-local spectra of height at most n (formally: E-local spectra, for E Morava E-theory). Pstrągowski [4] proved that if p > n^2 + n + 1, then this homotopy category is equivalent to the derived category of an abelian category. In other words, at large primes, this category is ‘algebraic’. These methods can be refined, as done by Patchkoria and Pstrągowski [3], to reduce this bound to 2(p-1) > n^2 + n (i.e. allowing for roughly twice as many primes). More recently, Barkan [2] further refined these constructions and showed that (if 2(p-1) > n^2 + 3n) this equivalence can be upgraded to a symmetric monoidal one; in other words, the smash product of such spectra is also algebraic.

In this talk I will present an introduction to these methods. First I will discuss Pstrągowski’s original approach, and then explain Barkan's way to upgrade it to a monoidal version. If time permits, I will briefly discuss Barkan’s work on arity approximations of ∞-operads [1], which his work relies on.

[1] S. Barkan, Arity approximation of ∞-operads.

[2] S. Barkan, Chromatic homotopy is monoidally algebraic at large primes.

[3] I. Patchkoria and P. Pstrągowski, Adams spectral sequences and Franke's algebraicity conjecture.

[4] P. Pstrągowski, Chromatic homotopy theory is algebraic when p > n^2 + n + 1.

The complete graph operad

Notes (sections 4 & 5)

Following work of Kashiwabara, Clemens Berger [1] introduced the so-called complete graph operads G_n (one for each n) and claimed these were E_n-operads. These operads played a central role in the theory, for example in the proof by McClure and Smith [2] of the Deligne conjecture. However, Berger's paper contains several mistakes, and the literature didn't seem to contain a correct proof that G_n actually is an E_n-operad.

I will make an attempt to fill this gap.

[1] C. Berger, Combinatorial models for real configuration spaces and En-operads.

[2] J. E. McClure and J. H. Smith, A solution of Deligne's Hochschild cohomology conjecture.

Goodwillie calculus and simplicial Lie algebras

The category of simplicial Lie algebras attracted some attention in the 60s and 70s due to its connection to the unstable Adams spectral sequence. It can be seen as a first approximation to the homotopy theory of spaces: many well-known phenomena from homotopy theory occur in a simpler and more regular form in this category. In a recent preprint [1], Nikolay Konovalov studies Goodwillie calculus in simplicial Lie algebras, and proves that the Goodwillie spectral sequence collapses at the E2 page for all spheres, which can be seen as an analogue of the Whitehead conjecture.

I will introduce the homotopy theory of simplicial Lie algebras and give an outline of Konovalov’s proof in my talk.

[1] N. Konovalov, Algebraic Goodwillie spectral sequence.

Homological stability for automorphism groups

A family of groups G_1 → G_2 → G_3 → ··· is said to satisfy homological stability if the induced maps H_i(G_n) → H_i(G_{n+1}) are isomorphisms in a range increasing with n. In [1] Randal-Williams and Wahl prove that homological stability always holds if there is a monoidal category (C, ⊕) satisfying a certain hypothesis and a pair of objects A and X in C, such that G_n is the group of automorphisms of A ⊕ X^{⊕n}. Their theorem applies to all the classical examples and gives new stability results in a unified way.

In this talk I will describe some of the tools that go into these homological stability results. At face value this talk will have no connection to calculus of functors, but if time permits, I'll allow myself to explain a potential connection.

[1] O. Randal-Williams and N. Wahl, Homological stability for automorphism groups.

E_3-multiplications on the truncated Brown-Peterson spectrum

The truncated Brown-Peterson spectrum BP<n> was originally constructed via its Postnikov tower so that its cohomology is isomorphic to A/(Q_0, …, Q_n), where A is the Steenrod algebra and Q_i is the i-th Milnor primitive. This was later shown by Quillen to admit a construction coming from complex cobordism and formal group laws. From the latter point of view, BP<n> was known to represent a multiplicative cohomology theory by geometric considerations due to Bass-Sullivan. Baker-Jeanneret used obstruction theory to show that this structure refines to an E_1-multiplication.

I will discuss recent work [1] of Hahn-Wilson, who show that there is a further refinement to an E_3-multiplication. They use this structure to prove redshift for BP<n>, namely that K(BP<n>) has chromatic height n+1.

[1] J. Hahn and D. Wilson, Redshift and multiplication for truncated Brown-Peterson spectra.

The Hilton-Milnor theorem

I will attempt to explain why the most basic version of the Hilton-Milnor theorem [3] «the homotopy groups of a wedge of two spheres look like a free Lie algebra» is true. Time permitting, I will also say something about interactions with Goodwillie calculus due to Arone-Kankaanrinta [1] and Brantner-Heuts [2].

[1] G. Arone and M. Kankaanrinta, The homology of certain subgroups of the symmetric group with coefficients in Lie(n).

[2] L. Brantner and G. Heuts, The vn-periodic Goodwillie tower on wedges and cofibres.

[3] P. J. Hilton, On the homotopy groups of the union of spheres.

Nilpotence and descent in equivariant stable homotopy theory

Ryan Quinn

In the 70's Quillen [4, 5] showed that the cohomology of a group is largely determined by the cohomology of its abelain subgroups. One of the key tools used was, what is now known as, Quillen's complex oriented descent. In the 90's work of Hopkins-Kuhn-Ravenel [2], and Greenlees-Strickland [1], extended Quillen's results to general complex oriented cohomology theories.

I will give an overview of the abstract nilpotence technology of Mathew-Naumann-Noel [3] that they used to further generalize these results.

[1] J. Greenlees and N. Strickland, Varieties and local cohomology for chromatic group cohomology rings.

[2] M. Hopkins, N. Kuhn and D. Ravenel, Generalized group characters and complex oriented cohomology theories.

[3] A. Mathew, N. Naumann and J. Noel, Nilpotence and descent in equivariant stable homotopy theory.

[4] D. Quillen, The spectrum of an equivariant cohomology Ring, I.

[5] D. Quillen, The spectrum of an equivariant cohomology Ring, II.

∞-Categorical formalism for equivariant higher algebra

Nardin and Shah [4] have recently developed an ∞-categorical approach to equivariant higher algebra. This is a suitable adaptation of Lurie’s work [3].

I will give a gentle introduction to this approach to equivariant higher algebra; the main highlight being the definition of G-equivariant ∞-operads and symmetric monoidal ∞-categories. We will see a slightly different but equivalent definition that is due to Barkan-Haugseng-Steinebrunner [1]. I will give a range of examples, in particular, I will explain how to write down N_∞-operads in this language. If time permits, I would like to discuss the equivariant version of the little disks operad and possibly sketch how it relates to equivariant factorization homology [2]. To follow this talk, no prior knowledge of Lurie’s model for (non-equivariant) ∞-operads is needed.

[1] S. Barkan, R. Haugseng and J. Steinebrunner, Enveloped for algebraic patterns.

[2] A. Horev, Genuine equivariant factorization homology.

[3] J. Lurie, Higher algebra.

[4] D. Nardin and J. Shah, Parametrized and equivariant higher algebra.

Christmas break

Happy holidays everyone!