  We derive a decoupling formula for the Brownian sheet which has the following
ready consequence: An $N$-parameter Brownian sheet in $\mathbf{R}^d$ has double
points if and only if $d<4N$. In particular, in the critical case where $d=4N$,
the Brownian sheet does not have double points. This answers an old problem in
the folklore of the subject. We also discuss some of the geometric consequences
of the mentioned decoupling, and establish a partial result concerning
$k$-multiple points in the critical case $k(d-2N)=d$.
