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

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

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]).

After this, we will decide which of the black boxes we want to fill in in the remaining time

### 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