  In this paper we study a spectrum $K(\mathcal{V}_k)$ such that $\pi_0
K(\mathcal{V}_k)$ is the Grothendieck ring of varieties and such that the
higher homotopy groups contain more geometric information about the geometry of
varieties. We use the topology of this spectrum to analyze the structure of
$K_0[\mathcal{V}_k]$ and show that classes in the kernel of multiplication by
$[\mathbb{A}^1]$ can always be represented as $[X]-[Y]$ where $X$ and $Y$ are
varieties such that $[X] \neq [Y]$, $X\times \mathbb{A}^1$ and $Y\times
\mathbb{A}^1$ are not piecewise isomorphic, but $[X\times \mathbb{A}^1]
=[Y\times \mathbb{A}^1]$ in $K_0[\mathcal{V}_k]$. Along the way we present new
proofs of the result of Larsen--Lunts on the structure on
$K_0[\mathcal{V}_k]/([\mathbb{A}^1])$.
