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

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 N between manifolds simultaneously off of a collection of pairwise disjoint submanifolds Q1,,Qj 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.

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intersection, linking, bordism, Blakers–Massey, functor calculus
Mathematical Subject Classification 2010
Primary: 55N45, 57N65, 55R70
Secondary: 57Q45, 57R40, 57R85, 57R91
Received: 23 November 2016
Revised: 22 August 2018
Accepted: 24 October 2018
Published: 21 May 2019
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