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We derive a decoupling formula for the Brownian sheet which has the followingready consequence: An $N$-parameter Brownian sheet in $\mathbf{R}^d$ has doublepoints 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 inthe folklore of the subject. We also discuss some of the geometric consequencesof the mentioned decoupling, and establish a partial result concerning$k$-multiple points in the critical case $k(d-2N)=d$.

Critical Brownian sheet does not have double points