$\mathbb A^1$-connectivity on Chow monoids v.s. rational equivalence of   algebraic cycles
  Let $k$ be a field of characteristic zero, and let $X$ be a projective variety embedded into a projective space over $k$. For two natural numbers $r$ and $d$ let $C_{r,d}(X)$ be the Chow scheme parametrizing effective cycles of dimension $r$ and degree $d$ on the variety $X$. An effective $r$-cycle of minimal degree on $X$ gives rise to a chain of embeddings of $C_{r,d}(X)$ into $C_{r,d+1}(X)$, whose colimit is the connective Chow monoid $C_r^{\infty }(X)$ of $r$-cycles on $X$. Let $BC_r^{\infty }(X)$ be the motivic classifying space of this monoid. In the paper we establish an isomorphism between the Chow group $CH_r(X)_0$ of degree $0$ dimension $r$ algebraic cycles modulo rational equivalence on $X$, and the group of sections of the sheaf of $\mathbb A^1$-path connected components of the loop space of $BC_r^{\infty }(X)$ at $Spec(k)$. Equivalently, $CH_r(X)_0$ is isomorphic to the group of sections of the $S^1\wedge \mathbb A^1$-fundamental group $\Pi _1^{S^1\wedge \mathbb A^1}(BC_r^{\infty }(X))$ at $Spec(k)$. 
Vladimir Guletskii
math.AG