This talk will describe a bordism theoretic obstruction to deforming a map between manifolds simultaneously off of a finite collection of pairwise disjoint submanifolds of the target under the hypothesis that the given map can be deformed off of any proper sub-collection of these submanifolds. In a certain range of dimensions, this will be the entire obstruction to finding the desired deformation. We apply this machine to give bordism type obstructions to lifting elements in the Goodwillie-Weiss embedding tower. We also give an application to the analysis of the homotopy type of the space of link maps.