Homotopical intersection theory, III: Multirelative intersection problems

Klein, John; Williams, Bruce

doi:10.2140/agt.2019.19.1079

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Homotopical intersection theory, III: Multirelative intersection problems

John R Klein and Bruce Williams

Algebraic & Geometric Topology 19 (2019) 1079–1134

DOI: 10.2140/agt.2019.19.1079

Abstract

We extend some results of Hatcher and Quinn (1974) beyond the metastable range. We give a bordism-theoretic obstruction $χ (f)$ 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, $χ (f)$ 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

References

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

Volume 19, issue 3 (2019)