Homotopy Theory: Revisiting Localizations and Beyond

Nathaniel Stapleton On the rationalization of the K(n)-local sphere
Since the 70s it has been expected that the homotopy groups of the rationalization of the K(n)-local sphere are an exterior algebra over the p-adic rationals on classes in degrees 1-2i for i from 1 to n. Recently, Barthel, Schlank, Weinstein and I have discovered a proof of this result by making use of a result of Faltings. In this talk, I will give some background on the K(n)-local sphere, explain the reduction to arithmetic geometry, and then give an idea of the modern techniques in arithmetic geometry that go into the proof.

Boris Chorny Small functors in homotopy theory
The idea to use small functors to generalize Elmendorf's construction of the Quillen equivalence between the equivariant model structure on spaces and the projective model structure on the presheaves over the diagram categories was suggested by Emanuel Dror Farjoun. In this talk we will show how this idea was implemented and discuss its further development and applications of small functors to Goodwillie calculus, representability theorems and more.

Jérôme Scherer Module spaces over homology theories and completion towers
This is joint work with W. Chacholski and W. Pitsch. To any coaugmented functor X=>EX one can associate its "modules", i.e. spaces X which are retracts of EX. When E is ordinary homology, this functor is the infinite symmetric product and modules are generalized Eilenberg-Mac Lane spaces (GEMs). Given two such functors E and F  we introduce a new one, we denote by a bracket [E, F], as a pullback of two natural transformations E=>EF and F=>EF. We study modules over such brackets and construct two towers by bracketing systematically either on the left or the right. For ordinary homology we obtain two well-known towers: the Bousfield-Kan completion tower and the modified one as introduced by Dror Farjoun. In this talk I also hope to indicate how this can be applied to understand the preservation of module structures by localization functors.

Aravind Asok Weak-cellular classes and nilpotent motivic spaces
The notion of nilpotent motivic space is a generalization to motivic homotopy theory of the notion of nilpotent space introduced by Dror-Farjoun. In practice, nilpotent motivic spaces can be studied just as easily as simply connected motivic spaces, but we will see that "geometrically natural" motivic spaces (e.g., the projective line) are frequently nilpotent and have non-trivial non-abelian fundamental sheaf of groups.  We will then introduce a notion of (weak) cellular classes in motivic homotopy theory; the latter theory is useful, for example, to formulate the motivic analog of the Freudenthal suspension theorem.  I will survey some joint work with subsets of Tom Bachmann, Jean Fasel and Mike Hopkins exploring these ideas.

Jean-Claude Hausmann The cell-dispensability obstruction for spaces and manifolds
We compare two properties for a CW-space X of finite type:
(1) being homotopy equivalent to a CW-complex of finite type without j-cells for k≤j≤ℓ ((k,ℓ)-cellfree) and
(2) H^j(X;R)=0 for any Zπ_1(X)-module R when k≤j≤ℓ (cohomogy (k,ℓ)-silent).
Using the technique of Wall's finiteness obstruction, we show that a connected CW-space X of finite type which is cohomogy (k,ℓ)-silent determines a "cell-dispensability obstruction" w_k(X) in K~0(Zπ_1(X)) which vanishes if and only if X is (k,ℓ)-cellfree (k≥4). Any class in K~0(Zπ) may occur as the cell-dispensability obstruction w_k(X) for a CW-space X with π_1(X) identified with π. Using projective surgery, similar results are obtained for manifolds, replacing "cells" by "handles" (antisimple manifolds). However, the realization statement is here only partial and open in general.

Bastiaan Cnossen Parametrized higher semiadditivity
Semiadditivity of a category, i.e. the existence of finite biproducts, provides useful algebraic structure on the category in the form of a canonical enrichment in abelian monoids. It was shown by Harpaz that this generalizes to the context of higher semiadditivity introduced by Hopkins and Lurie for applications in chromatic homotopy theory. In this talk, I will explain a further generalization to the context of ambidexterity for families​ of categories, allowing for applications in equivariant homotopy theory, six-functor formalisms and transchromatic character theory. This is joint work with Tobias Lenz and Sil Linskens.

Wojciech Chachólski Vietoris Rips complexes, Cellularity, and Topological Data Analysis (TDA).
The ability to extract global information from local is important. What is meant by local information however depends on the input and the description of considered spaces. For example what is often understood as local information in TDA differs from the local information typically considered in topology. In this talk I will discuss how to estimate the difference between these two notions of local information using cellularity. I will also take this opportunity to illustrate how homology can be used to analyse data. 

Gijs Heuts Periodic localizations of homotopy theory and Hopf algebras
I will discuss the relation between localizations of homotopy theory at v_n-periodic homotopy equivalences and T(n)-homology equivalences. The first can be understood through Lie algebras in spectra, whereas the second turns out to be closely related to Hopf algebras in spectra. I will explain this perspective and state several open questions. Much of this is based on ongoing joint work with Brantner, Hahn, and Yuan.

Carles Casacuberta Consistency strength of homotopical localizations
Farjoun asked in 1992 if every continuous homotopy idempotent functor on spaces can be obtained by inverting a single map. It has been known since 2010 that this claim is equivalent to a large-cardinal axiom known as Vopenka principle. We prove that the existence of localizations of spaces or spectra at arbitrary classes of maps is equivalent to a much weaker large-cardinal axiom, which lies below supercompactness. The precise consistency strength of the existence of cohomological localizations remains unknown.

Michael Hopkins TBA

Ayelet Lindenstrauss Loday constructions on spheres and tori
(All joint work with Birgit Richter, and parts also with others as detailed during the talk)
Given a commutative ring or ring spectrum, there is a functor from finite sets to commutative rings sending a set to the tensor product of copies of the ring (or ring spectrum) indexed by that set. The Loday construction extends this to simplicial sets. The simplest nontrivial example is the Loday construction over a circle, which gives Hochschild homology (or, respectively, topological Hochschild homology). I will discuss Loday constructions on higher spheres and on higher tori. The calculations for spheres are inductive, using the splitting of an n-sphere into two hemispheres, joined along the equator (n-1)-sphere. The calculations for tori which we know are for cases where the Loday construction on the torus is equivalent to the Loday construction on the wedge of spheres whose suspension is homotopy equivalent to the suspension of the torus. This kind of stability is unexpected, definitely not generally true, and yet there are surprisingly many cases where it holds.

Shaul Ragimov Semiadditive alternating powers
We investigate representations of the symmetric group derived from characters, particularly within the higher semiadditive context. This leads to a generalized m-th "alternating power", encompassing classical symmetric and alternating powers, as well as symmetric and exterior powers of categories as described in [arXiv:1110.4753]. We introduce a notion of "twisted power operations" and demonstrate that in algebraically closed examples of interest, these two notions are related by dimension, modulo a height-dependent shift. This yields an induction formula for the dimension of an alternating power extending [arxiv:2310.00275]. For categories of height n with a cyclotomically closed unit, any element in the (n+1) homotopy group of the sphere yields a character of the symmetric group. These resulting alternating powers form a commutative algebra in the category of 𝐙-graded objects endowed with a "Koszul" symmetric monoidal structure. Particularly in height 0, this reproduces the exterior algebra as a graded-commutative algebra. We leverage this to establish connections between generating functions of dimensions of alternating powers corresponding to different elements in the homotopy group in low dimensions. We also give complete computation of dimensions in height smaller or equal to 2. This is a joint work with Shai Keidar.

Shaul Barkan Chromatic homotopy is monoidally algebraic for p>O(h^2)
Stable homotopy theory is intimately related to the geometry of formal groups through the Adams Novkiov spectral sequence. Franke made this analogy precise by introducing a derived category of certain sheaves on the moduli stack of formal groups as a candidate algebraic model for the infinity category of spectra of chromaric height ≤h. He conjectured that at primes sufficiently larger than the height the homotopy truncation of the two categories coincide. Barthel-Schlank-Stapleton proved an asymptotic variant of Franke’s conjecture using categorical ultraproducts. Later, Pstragowski proved an effective yet non-monoidal version of the conjecture. In the talk I will explain the effective solution for the symmetric monoidal formulation of Franke’s conjecture.

Matthias Kreck Betti numbers of closed manifolds
This is joint work with Don Zagier. We study the following basic question: Given natural numbers b_0, b_1,…., b_n, is there a closed smooth n-dimensional manifolds M with Betti numbers b_i? One has to distinguish two cases: non-orientable manifolds and orientable manifolds. In the first case we have a complete answer and the proof is simple. In the orientable case we have a complete translation into algebra and number theory in which rational homotopy plays a central role. Does this answer the question? No, and part of our message is, that we conjecture that there will never be a complete answer. On the way some crazy phenomena will be explained.