Seminar zur Motiven SS 2026

Motives Seminar-Invited Lectures

This semester the motives seminar will be a series of invited talks. We meet in WSC-N-U-4.04, 16-18 Uhr (c.t.). Talks will also be available via Zoom; the Zoom link will be in the lecture announcement. If you are not on our mailing list and would like the link, please send me a request at marc.levine@uni-due.de, similarly if you are interested in giving a talk.

Schedule

April 14, Xiaowen Dong (Essen). Unstable relations for motivic Hopf maps and motivic Toda brackets

Abstract In classical topology we have the famous Hopf maps $\eta_{top}: S^3\rightarrow S^2$ and $\nu_{top}:S^7\rightarrow S^4$. The first Hopf map $\eta_{top}$ is the canonical map from the unit sphere in $\mathbb{C}^2$ to the complex projective space $\mathbb{C}\mathbb{P}^1$. Instead of complex numbers we can also use quaternions and get the second Hopf map $\nu_{top}$ in the same way. Furthermore if we consider the stabilizations of both Hopf maps, we have the relations $2\eta_{top}=0$ and $\eta_{top}\nu_{top}=0$ in classical stable homotopy theory.

In motivic homotopy theory we have analogues of the topological Hopf maps. For this talk we work over the base scheme $\mathrm{Spec} \ \mathbb{Z}$. The first motivic Hopf map $\eta$ is defined to be the canonical projection from $\mathbb{A}_{\mathbb{Z}}^2-\{0\}$ to $\mathbb{P}_{\mathbb{Z}}^1$. Moreover, this map can also be constructed via the Hopf construction on the scheme $\mathbb{G}_{m}$. The Hopf construction on $\mathrm{SL}_{2}$ gives us the second motivic Hopf map $\nu$. The complex realizations of $\eta$ and $\nu$ are the corresponding topological Hopf maps.

One can consider the $\mathbb{P}^1$-stabilizations of $\eta$ and $\nu$ in the motivic stable homotopy category SH$(\mathbb{Z})$. One also has the endomorphism $h:=\langle 1\rangle+\langle -1\rangle$ of the unit $\mathbb{S}_{\mathbb{Z}}$ in SH$(\mathbb{Z})$, with complex realisation $2\in\pi_{0}(\mathbb{S})=\mathbb{Z}$; here the endomorphisms $\langle \pm 1\rangle$ of $\mathbb{S}_{\mathbb{Z}}$ are the $\mathbb{P}^1$-stabilizations of the automorphisms $[x_0:x_1]\mapsto [x_0:\pm x_1]$ of $\mathbb{P}_{\mathbb{Z}}^1$. Dugger, Isaksen and Morel prove the relations $h\eta=\eta h=0$ and $\eta\nu=\nu\eta=0$ in SH$(\mathbb{Z})$.

In this talk, we will discuss versions of these relations in the motivic unstable homotopy category $\mathcal{H}(\mathbb{Z})$, and show how one obtains non-trivial examples of Toda brackets from these nullhomotopic relations.

April 21, Sil Linskens (University. Regensburg). t.b.a.

April 28, Svetlana Makarova (University. Melbourne). t.b.a.

May 5, N.N.

May 12, N.N.

June 23, N.N.

July 7, N.N.

July 14, N.N.

July 21, N.N.