${\mathbb{A}}^1$-fibration in algebraic geometry and ${\mathbb{A}}^1$-homotopy type

Abstract

In this article we show that an ${\mathbb{A}}^1$ bundle map or a vector bundle map $p:X\to Y$ induces trivial local fibration $\underline{Sing}(X)\to \underline{Sing}(Y)$. Using this, we first show that for Korus Russel threefolds of first kind $X$ the space $\underline{Sing}(X)$ is ${\mathbb{A}}^1$ local. Then we show that for any smooth affine complex surface $X$, the ${\mathbb{A}}^1$- connected component sheaf is homotopy invariant.