  This paper extends some results of Hatcher and Quinn beyond the metastable
range. We give a bordism theoretic obstruction to deforming a map between
manifolds simultaneously off of a collection of pairwise disjoint submanifolds
under the assumption that it can be deformed off of any proper subcollection in
a homotopy coherent way. In a certain range of dimensions, ours is a complete
obstruction to finding the desired deformation. We apply this machinery to
embedding problems and to the study of linking phenomena.
