On the unstable intersection conjecture

Levin, Michael

doi:10.2140/gt.2018.22.2511

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On the unstable intersection conjecture

Michael Levin

Geometry & Topology 22 (2018) 2511–2532

DOI: 10.2140/gt.2018.22.2511

Abstract

Compacta $X$ and $Y$ are said to admit a stable intersection in $ℝ^{n}$ if there are maps $f : X \to ℝ^{n}$ and $g : Y \to ℝ^{n}$ such that for every sufficiently close continuous approximations $f^{'} : X \to ℝ^{n}$ and $g^{'} : Y \to ℝ^{n}$ of $f$ and $g$ , we have $f^{'} (X) \cap g^{'} (Y) \neq \emptyset$ . The unstable intersection conjecture asserts that $X$ and $Y$ do not admit a stable intersection in $ℝ^{n}$ if and only if $dim X \times Y \leq n - 1$ . This conjecture was intensively studied and confirmed in many cases. we prove the unstable intersection conjecture in all the remaining cases except the case $dim X = dim Y = 3$ , $dim X \times Y = 4$ and $n = 5$ , which still remains open.

Keywords

cohomological dimension, extension theory

Mathematical Subject Classification 2010

Primary: 55M10

Secondary: 54F45, 55N45

References

Publication

Received: 19 November 2015

Revised: 19 October 2017

Accepted: 1 November 2017

Published: 1 June 2018

Proposed: Steve Ferry

Seconded: Peter Teichner, David Gabai

Authors

Michael Levin
Department of Mathematics Ben Gurion University of the Negev Be’er Sheva Israel

Volume 22, issue 5 (2018)