#### Volume 22, issue 5 (2018)

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On the unstable intersection conjecture

### Michael Levin

Geometry & Topology 22 (2018) 2511–2532
##### 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 ${f}^{\prime }:X\to {ℝ}^{n}$ and ${g}^{\prime }:Y\to {ℝ}^{n}$ of $f$ and $g$, we have ${f}^{\prime }\left(X\right)\cap {g}^{\prime }\left(Y\right)\ne \varnothing$. The unstable intersection conjecture asserts that $X$ and $Y$ do not admit a stable intersection in ${ℝ}^{n}$ if and only if $dimX×Y\le n-1$. This conjecture was intensively studied and confirmed in many cases. we prove the unstable intersection conjecture in all the remaining cases except the case $dimX=dimY=3$, $dimX×Y=4$ and $n=5$, which still remains open.

##### Keywords
cohomological dimension, extension theory
##### Mathematical Subject Classification 2010
Primary: 55M10
Secondary: 54F45, 55N45