Stable $\A^1$-connectedness

Abstract

We prove in this note a stabilized version of a conjecture on $\A^1$-connectedness. For the stabilized version of this conjecture, we introduce the notion of stable $\A^1$-connectedness, which is can be seen as the stabilization of $\A^1$-connectedness. The notion of stable $\A^1$-connectedness is in connection to the existence of zero cycles of degree one rather than rational points.