  Perron and Quinn gave independent proofs in 1986 that every topological
pseudo-isotopy of a simply-connected, compact topological 4-manifold is
isotopic to the identity. Another result of Quinn is that every smooth
pseudo-isotopy of a simply-connected, compact, smooth 4-manifold is smoothly
stably isotopic to the identity. From this he deduced that
$\pi_4(\operatorname{TOP}(4)/\operatorname{O}(4)) =0$. A replacement criterion
is used at a key juncture in Quinn's proofs, but the justification given for it
is incorrect. We provide different arguments that bypass the replacement
criterion, thus completing Quinn's proofs of both the topological and the
stable smooth pseudo-isotopy theorems. We discuss the replacement criterion and
state it as an open problem.
