Connectivity of motivic \(H\)-spaces (Q384638)

Let \(Sm/k\) be the category of smooth separated \(k\)-schemes, \(PSH(Sm/k)\) the category of presheaves of sets on \(Sm/k\). Let \(\Delta\) be the category of simplices and \(\Delta^{\mathrm{op}}PSh(Sm/k)\) the category of spaces. A functor \({\mathcal X}:\Delta^{\mathrm{op}}\to PSh(Sm/k)\) is called a simplicial presheaf or a space. \(\Delta^{\mathrm{op}}PSh(Sm/k)\) has a local model category structure with respect to to the Nisnevich topology. The Bousfield localization of the local model structure on \(\Delta^{\mathrm{op}}PSh(Sm/k)\), with respect to the class of maps \({\mathcal X}\times\mathbb{A}^1\to{\mathcal X}\), is called the \(\mathbb{A}^1\)-model structure. Let \(\mathbb{H}(k)\) be the resulting homotopy category. For any space \({\mathcal X}\), \(\pi^{\mathbb{A}^1}_0({\mathcal X})\) is the presheaf NEWLINE\[NEWLINEU\in Sm/k\to \Hom_{\mathbb{H}(k)}(U,{\mathcal X})NEWLINE\]NEWLINE which is homotopy invariant, i.e. for any \(U\in Sm/k\) the morphism NEWLINE\[NEWLINE\pi^{\mathbb{A}^1}_0({\mathcal X})(U)\to \pi^{\mathbb{A}^1}_0({\mathcal X})(\mathbb{A}^1_U)NEWLINE\]NEWLINE induced by the projection \(\mathbb{A}^1_U\to U\), is bijective. Let NEWLINE\[NEWLINE\alpha_{\mathrm{Nis}}: PSh(Sm/k)\to Sh_{\mathrm{Nis}}(Sm/k)NEWLINE\]NEWLINE denote the Nisnevich sheafification functor. A conjecture by Morel states that, for any \(U\in Sm/k\), the morphism NEWLINE\[NEWLINE\alpha_{\mathrm{Nis}}(\pi^{\mathbb{A}^1}_0({\mathcal X})(U))\to \alpha_{\mathrm{Nis}}(\pi^{\mathbb{A}^1}_0({\mathcal X})(\mathbb{A}^1_U))NEWLINE\]NEWLINE is bijective.NEWLINENEWLINE In this paper the author proves that the \(\mathbb{A}^1\)-connected component sheaf \(\alpha_{\mathrm{Nis}}(\pi^{\mathbb{A}^1}_0({\mathcal X}))\) of an \(H\)-group is \(\mathbb{A}^1\)-invariant. The following result gives a partial answer to the conjecture.NEWLINENEWLINE Theorem 1. For any space \({\mathcal X}\), the canonical morphism NEWLINE\[NEWLINE\pi_0({\mathcal X})(\mathbb{A}^1_F)\to \alpha_{\mathrm{Nis}}(\pi^{\mathbb{A}^1}_0({\mathcal X})(\mathbb{A}^1_F))NEWLINE\]NEWLINE is bijective for all finitely generated separable field extensions \(F/k\).