Homotopical Intersection Theory, III: multi-relative intersection problems

TL;DR

This paper develops a bordism-based obstruction theory for deforming maps off multiple submanifolds simultaneously, extending previous results and applying to embedding and linking problems in topology.

Contribution

It introduces a new obstruction framework that generalizes prior work to multi-relative intersection problems beyond the metastable range.

Findings

Provides a complete obstruction in certain dimension ranges

Extends homotopical intersection theory beyond previous limits

Applies to embedding and linking phenomena in manifolds

Abstract

This paper extends some results of Hatcher and Quinn 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 it can be deformed off of any proper subcollection in a homotopy coherent way. In a certain range of dimensions, ours is a complete obstruction to finding the desired deformation. We apply this machinery to embedding problems and to the study of linking phenomena.