#### Volume 19, issue 3 (2019)

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Homotopical intersection theory, III: Multirelative intersection problems

### John R Klein and Bruce Williams

Algebraic & Geometric Topology 19 (2019) 1079–1134
##### 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.

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##### Keywords
intersection, linking, bordism, Blakers–Massey, functor calculus
##### Mathematical Subject Classification 2010
Primary: 55N45, 57N65, 55R70
Secondary: 57Q45, 57R40, 57R85, 57R91