We extend some results of Hatcher and Quinn (1974) 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 can be deformed off of any proper subcollection in a homotopy coherent way. In a certain range of dimensions, is a complete obstruction to finding the desired deformation. We apply this machinery to embedding problems and to the study of linking phenomena.
"Homotopical intersection theory, III: Multirelative intersection problems." Algebr. Geom. Topol. 19 (3) 1079 - 1134, 2019. https://doi.org/10.2140/agt.2019.19.1079