# Motives Seminar

## The motivic Freudenthal suspension theorem, following Asok, Bachmann and Hopkins

The classical Freundenthal suspension theorem asserts that for $n\ge2$ and $X$ a pointed, $n-1$-connected CW complex, the unit map $X\to \Omega\Sigma X$ is $2n − 1$-connected, i.e., an isomorphism on homotopy groups in degrees $\le 2n − 2$ and an epimorphism in degree $2n − 1$. Since $S^n$ is $n-1$-connected, one can iterate to recover results on $\pi_{n+i}(S^n)$ from the $i$th stable homotopy group of the sphere spectrum $\pi_i^s(1):=\text{colim}_{m\to\infty} \pi_{i+m}S^m,$ for $i\le n-2$. The stable groups are in general easier to compute (at least in low degree!). In the motivic world, one has a similar result for the $S^1$-stable theory, due to Morel, Asok-Fasel, Wickelgren-Williams, and others, however, for geometric applications, it is really necessasy to understand the $\mathbb{P}^1$-stable version. This is contained in the following theorem of Asok-Bachmann-Hopkins

Theorem We work over a field $k$ of characteristic zero. Let $X$ be a pointed motivic space that is weakly $S^{p,q}$-cellular, with $p-q\ge2$ and $q\ge2$. Then the fiber of the unit map $X\to \Omega_{\mathbb{P}^1}\Sigma_{\mathbb{P}^1}X$ is weakly $S^{a,2q}$-cellular, where $a=\text{min}(2p-1, p+2q-1)$.

Here the weakly cellular'' condition replaces the classical notion of connectivity, and is defined as follows: $X$ is weakly $S^{p,q}$-cellular if $X$ is in the smallest subcategory of motivic spaces containing the objects $S^{p,q}\wedge Y_+$ with $Y$ a smooth finite type $k$-scheme, and closed under weak equivalences, (small) colimits and extensions in homotopy cofiber sequences.

We will meet on Tuesdays, 16:15-18 Uhr, in WSC-S-U-3.03.

### Program

Lecture 1. April 9: Marc Levine.  An overview
Lecture 1 Video

Lecture 2. April 16:  Chirantan Chowdhury.  §2: Motivic localization and group theory, §3 [ABH22]: Connectivity of fibers and cofibers
Lecture 2 Video
For [§2.1, ABH22], you will need some results from [AFH22]: Lemma 3.1.14, Def. 2.1.3, discussion of the $i^{th}$ highest center on pg. 675, Prop. 3.1.22 and Def. 3.2.1. Note that the references to [AFH22, Lemma 3.1.14] are sometimes misstated as Lemma 3.1.4.

Lecture 3: April 23:  Svetlana Makarova.  A resumé of results from [AFH22], part 1: §2.1 (Nilpotent (pre)sheaves of groups), §2.2 (Nilpotence in local homotopy theory), §3.1 ($\mathbb{A}^1$-local group theory).
Lecture 3 Video
Be sure to cover Def. 2.1.1, Def. 2.1.3 (if not already covered in Lecture 2), Def. 3.1.5, Thm. 3.1.8, Thm. 3.1.12, Lemma 3.1.14 (if not already covered in Lecture 2), Def. 3.1.17, Prop. 3.1.18, Thm. 3.1.19, Prop. 3.1.22 (if not already covered in Lecture 2).

Lecture 4: April 30:  Marc Levine.  A resumé of results from [AFH22], part 2: §3.2 ($\mathbb{A}^1$-nilpotent groups), §3.3 ($\mathbb{A}^1$-nilpotent spaces and $\mathbb{A}^1$-fiber sequences), §3.4 (Examples), § 4.1 (Sheaf cohomology and $\mathbb{A}^1$-homology).
Lecture 4 Video
Be sure to cover Def. 3.2.1 (if not already covered in Lecture 2), Prop. 3.2.3, Lemma 3.2.7,
Def. 3.3.1, Prop. 3.3.2, Thm. 3.3.6, Def. 3.3.9, Def. 3.3.11, The. 3.3.13, Thm. 3.3.17
Ex. 3.4.1, The. 3.4.8.
Notions of sheaf cohomology $\mathbb{A}^1$-homology pgs. 690, 691, Prop. 4.1.2, Lemma 4.1.4.

Lecture 5. May 7:  Herman Rohrbach §4 [ABH22]: Abelianization and $\mathbb{A}^1$-lower central series
Lecture 5 Video

Lecture 6. May 14:  Thomas Brazelton  §5 [ABH22]: Principal refinements of Moore-Postnikov factorizations, §6 [ABH22]: Applications. In section 6, we just need through Prop. 6.2.
Lecture 6 Video

Lecture 7. May 21:  Clémentine Lemaire-Rieusset. §2 [ABH23]: Preliminaries on unstable and stable motivic homotopy theory.
Lecture 7 Video .
Lecture 7 Slides

Lecture 8. May 28:  Pietro Gigli.  §3 [ABH23]: Weak cellularity and nullity
Lecture 8 Video.

Lecture 9. June 4:  Jan Hennig.   §4 [ABH23]: A weakly-cellular Whitehead tower and consequences
Lecture 9 Video

Lecture 10. June 11:  Herman Rohrbach.  §5 [ABH23]: Equivariant geometry of symmetric powers
Lecture 10 Video

Lecture 11. June 18:  Fei Ren.  §6 [ABH23]: Weak cellular estimates of the fiber of the unit map.
If time permits say something about the applications to Murthy's conjecture (§7 [ABH23]).
Lecture 11 Video

### References

[ABH22] A. Asok, T. Bachmann, M. Hopkins, On the Whitehead theorem for nilpotent motivic spaces, arXiv:2210.05933

[ABH23] A. Asok, T. Bachmann, M. Hopkins, On $\mathbb{P}^1$-stabilization in unstable motivic homotopy theory, arXiv:2306.04631

[AFH22] A. Asok, J. Fasel, M. Hopkins, Localization and nilpotent spaces in $\mathbb{A}^1$-homotopy theory, arXiv:1909.05185

### Special Lectures

June 25. A special lecture by Vijaylaxmi Trivedi (TIFR and SUNY Buffalo).
Title: Density functions for Epsilon multiplicity and families of ideals
Abstract: A density function for an algebraic invariant is a measurable function on $\mathbb{R}$ which measures the invariant on an $\mathbb{R}$-scale, that is, a density function for a given algebraic invariant, say $\epsilon$, is an integrable function $f_\epsilon\colon \mathbb{R}\to \mathbb{R}$, which gives a new measure $\mu_{f_\epsilon}$ on $\mathbb{R}$ such that the integration $\int_Ef_\epsilon$ on a subset $E\subset\mathbb{R}$ is `the measure $\mu_{f_\epsilon}$ of the invariant $\epsilon$ on $E$’.
This function, when it exists, seems to carry more information related to the invariant without seeking extra data (than needed to study the invariant itself). The HK density function, which was introduced by the speaker to study Hilbert-Kunz multiplicity, turned out to be a useful tool in answering two open questions.
In this talk we discuss the existence and applications of density functions for epsilon multiplicity (a notion introduced by Kleiman-Ulrich-Validashti to study integral closure of an ideal) and for some families of ideals. The talk is based on a joint work with Suprajo Das and Sudeshna Roy.
Lecture Video

July 2. A special lecture by Vova Sosnilo (Univ. Regensburg)
Title: Atiyah-Segal completion theorem for singular varieties.
Abstract: We formulate and prove a version of the AS completion theorem for singular varieties in characteristic 0. Along the way we discuss the cyclotomic trace techniques and the geometry of loop stacks. As an application, we construct a complete motivic filtration on the AS completion of the equivariant K-theory, extending Elmanto-Morrow filtration from the case of schemes. This is based on joint work in progress with Elden Elmanto and Dmitry Kubrak.