# 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.

### Previous Motives Seminars

Follow the ** Teaching** link at the top of this page to find links to previous motives seminars.

### 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

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).

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).

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: N.N. §4 [ABH22]: Abelianization and $\mathbb{A}^1$-lower central series

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

Lecture 7. May 21: N.N. §2 [ABH23]: Preliminaries on unstable and stable motivic homotopy theory

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: N.N. §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