Ivan Panin (St Petersburg/Oslo): Around the Voevodsky' conjecture.

Voevodsky's notion of rational  framed correspondences eventually will be recalled.
Voevodsky's conjecture will be formulated once again.

Conjecture. If k is an infinite perfect field, then for  each k-smooth variety X the canonical motivic space morphism
      \(\operatorname{Fr}^{rat}(\Delta^\bullet \times -, X) \rightarrow \Omega^\infty_{\mathbb{P^1}} \Sigma^\infty_{\mathbb{P}^1}(X_+)\)
is a Nisnevich local equivalence.

We recall notion of stable linear and stable linear rational framed correspondences from \(U\) to \(X\).
Let \(\operatorname{ZF}(U,X)\) and \(\operatorname{ZF}^{rational}(U,X) \)stand for  stable linear and stable linear rational framed correspondences from \(U\) to \(X\) respectively.
Using joint results of G.Garkusha and the speaker  and the linearization theorem proven by G.Garkusha, A.Neshitov and the speaker
one can prove that the Voevodsky conjecture is equivalent to the following one

Conjecture II. for each \(k\)-smooth variety \(X\) and each field \(K/k\)  the inclusion of complexes of abelian groups
               \( \operatorname{in}: \operatorname{ZF}(\Delta^\bullet_K, X) \rightarrow \operatorname{ZF}^{rational}(\Delta^\bullet_K, X)\)
is a quasi-isomorphism of complexes of abelian groups.

Take \(X=\operatorname{Spec}(k)\) and write \(pt\) for \(\operatorname{Spec}(k)\).  The latter conjecture yields an isomomorphism
\( i_*: H_0(\operatorname{ZF}(\Delta^\bullet_K, pt)) \rightarrow H_0(\operatorname{ZF}^{rational}(\Delta^\bullet_K, pt))\)
of the abelian groups. So, there is a very reasonable task: to check that the map \(i_*\) is an isomorphism.

This would be a very good test for the Conjecture II  to be true. It \(\operatorname{char}(k)=0\), then it is known due to A.Neshitov that the left hand side is
the Grothendieck-Witt group  \(\operatorname{GW}(K) \)of the field \(K\). So, if  \(i_*\) is an isomorphism, then \(H_0(ZF^{rational}(\Delta^\bullet_K, pt)) = \operatorname{GW}(K)\).