Homotopical intersection theory, III: Multirelative intersection problems

Abstract

We extend some results of Hatcher and Quinn (1974) beyond the metastable range. We give a bordism-theoretic obstruction $\chi \left(f\right)$ to deforming a map $f:P\to N$ between manifolds simultaneously off of a collection of pairwise disjoint submanifolds $Q_1,\dots ,Q_j\subset N$ under the assumption that $f$ can be deformed off of any proper subcollection in a homotopy coherent way. In a certain range of dimensions, $\chi \left(f\right)$ is a complete obstruction to finding the desired deformation. We apply this machinery to embedding problems and to the study of linking phenomena.