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.

Keywords
intersection, linking, bordism, Blakers–Massey, functor calculus
Mathematical Subject Classification 2010
Primary: 55N45, 57N65, 55R70
Secondary: 57Q45, 57R40, 57R85, 57R91
Publication
Received: 23 November 2016
Revised: 22 August 2018
Accepted: 24 October 2018
Published: 21 May 2019
Authors
 John R Klein Department of Mathematics Wayne State University Detroit, MI United States Bruce Williams Department of Mathematics University of Notre Dame Notre Dame, IN United States