  We prove a weak factorization result on birational maps of Deligne-Mumford
stacks, and deduce the following: Let $U \subset X$ be an open embedding of
smooth Deligne-Mumford stacks such that $D = X-U$ is a normal crossings
divisor, then the the simple homotopy type of the boundary complex
$\Delta(X,D)$ depends only on $U$.
