18 problems

Let $\mathcal{Y}$ be a space over a field, and let $\pi_i^{\mathbb{A}^1}(\mathcal{Y})$ denote its $i$-th $\mathbb{A}^1$-homotopy sheaf. Morel proved that…

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 $X$ be a smooth affine variety that is topologically contractible and $H\mathbb{Z}$-acyclic, and let $i\geq 2$. Write $\Sigma^i_{\mathbb{P}^1}X$ for its $i$-fold $\mathbb{P}^1$…

Motivic vector bundle conjecture. The answer to the question whether every topological vector bundle is motivic should always be yes. In particular, on $\mathbb{P}^n$, every topolo…

Let ${\mathbf H}^{{\mathbb A}^1}_n(X)$ denote the $n$th ${\mathbb A}^1$-homology sheaf of a motivic space $X$, and let $\mathbb G_m$ be the multiplicative group over the base field…

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 $X$ be a topologically contractible smooth affine complex variety. A smooth affine variety is stably $\mathbb{A}^{1}$-contractible when it becomes $\mathbb{A}^{1}$-contractible…

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…

Asok and Morel's conjecture. This natural map is an isomorphism for every proper scheme $X$ of finite type over a field.

Asok and Morel's conjecture. This natural epimorphism is an isomorphism for every smooth scheme $X$ over $k$.

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,…

Let $k$ be a field, let ${\mathcal Sm}_k$ denote the category of smooth $k$-schemes, and let $X\in{\mathcal Sm}_k$. Write $\pi_0^{ch}(X)$ for the sheaf of ${\mathbb A}^1$-chain con…