  We prove analogs of Whitehead's theorem (from algebraic topology) for both
the Chow groups and for the Grothendieck group of coherent sheaves: a morphism
between smooth projective varieties whose pushforward is an isomorphism on the
Chow groups, or on the Grothendieck group of coherent sheaves, is an
isomorphism. As a corollary, we show that there are no nontrivial naive
$\mathbb{A}^1$-homotopy equivalences between smooth projective varieties.
