E. Brown showed that for any
map f of a punctured disc Bn with n holes into a 2-manifold M that is an
embedding of ∂Bn, there is an embedding g of a punctured disk Bk into M such that
g(∂Bk) is a subcollection of f(∂Bn). In this paper E. Brown’s approach is extended
to show that a similar result holds for maps of punctured q-balls into certain
q-manifolds (q ≧ 3).