  We study the algebraic structure of the $n$-dimensional Cremona group and
show that it is not an algebraic group of infinite dimension (ind-group) if
$n\ge 2$. We describe the obstruction to this, which is of a topological
nature.
  By contrast, we show the existence of a Euclidean topology on the Cremona
group which extends that of its classical subgroups and makes it a topological
group.
