11 problems

Abelianness and strict invariance conjecture. For every reductive group $G$ over $k$, the sheaf $\pi_0^{{\mathbb A}^1}(G)$ is a sheaf of abelian groups. Moreover,…

Power-structure compatibility conjecture. For every such $k$, $X$, and $n$,

Asok--Fasel's exactness conjecture. The sequence

Let $k$ be a field, let $\mathcal{X}$ be a space, and write $\pi_0^{\mathbb{A}^1}(\mathcal{X})$ for its sheaf of $\mathbb{A}^1$-connected components. Morel's conjecture. The sheaf…

Let $G$ be an ${\mathbb A}^1$-connected, isotropic, semisimple, almost simple, simply connected algebraic group over $k$, split by a finite separable field extension $F/k$. Let…

Let $k$ be a perfect field with ${\rm char}(k)\ne 2$, and let $d$ be the dimension of a $k$-scheme $X$. Write $\mathbf{K}^{\mathrm{M}}_{d+2}/24$ for the quotient of the Milnor $K$-…

Let $X$ be a topologically contractible smooth complex affine variety, and let $x\in X({\mathbb C})$ be a chosen base-point. Write $\Sigma^n_{{\mathbb P}^1}(X,x)$ for the $n$-fold…

Let $X$ be a smooth topologically contractible surface. Surface contractibility conjecture. The variety $X$ is ${\mathbb A}^1$-contractible if and only if it is isomorphic to…

Let $X$ be a topologically contractible smooth complex variety with a chosen base-point $x\in X({\mathbb C})$. Write $\Sigma^n_{{\mathbb P}^1}(X,x)$ for the $n$-fold…

Let $k$ be a base field, let $n$ and $i$ be integers with $n\geq 4$ and $i\geq 0$, and write $S^{(n-1+i)+n\alpha}$ for the corresponding motivic sphere. Let…

Birationality and ${\mathbb A}^1$-invariance conjecture. For any proper scheme $X$ of finite type over a field $k$, this canonical morphism is an isomorphism; equivalently,…