Ivan Panin (St Petersburg/Oslo): Motivic delooping machinery

Title of the first lecture: Rational framed correspondences and a conjecture of Voevodsky

Abstract. In the first lecture we recall Voevodsky’s theory of framed correspondences, rational framed correspondences,
stable framed and stable rational framed correspondences. The latter two are denoted by \(\operatorname{Fr}(U,X)\) and \(\operatorname{Fr}^{\operatorname{rat}}(U,X)\) respectively.
Basing on a conjecture of Voevodsky, the homotopy groups of the simplicial set
\(\operatorname{Fr}^{\operatorname{rat}}(\Delta_{\mathbb{C}}^{\bullet} , \operatorname{Spec} (\mathbb{C}))\)
coincide with the stable homotopy groups of the topological sphere S^0, where C is the field of complex numbers.
More precisely, the above simplicial set has conjecturally the homotopy type of the classical topological space
\(\Omega^{\infty}\Sigma^{\infty}(S^0)\).
The Voevodsky conjecture asserts that for each perfect field k and each k-smooth variety X the canonical morphism of motivic spaces
\(\operatorname{Fr}^{\operatorname{rat}}(\Delta^{\bullet} \times -, X)\rightarrow \Omega^{\infty}_{\mathbb{P}^1} \Sigma^{\infty}_{\mathbb{P}^1}(X_+)\)
is a Nisnevich local equivalence. Using a theorem of Garkusha and Panin this conjecture is equivalent to saying that
\(\operatorname{Fr}^{\operatorname{rat}}(\Delta^{\bullet} \times -, X) \rightarrow \Omega^{\infty}_{\mathbb{P}^1} \Sigma^{\infty}_{\mathbb{P}^1}(X_+)\) is locally a group completion of the space \(\operatorname{Fr}(\Delta^{\bullet} \times -, X)\).

In other lectures we will discuss motivic delooping machinery developed by G. Garkusha and I. Panin in [2] based on Voevodsky's theory of framed correspondences [1].

[1] V. Voevodsky. Notes on framed correspondences (his home page).
[2] G. Garkusha, I. Panin, Framed motives of algebraic varieties (after V. Voevodsky), preprint arXiv:1409.4372v4