On the unstable intersection conjecture

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 $dimX\times Y\le n-1$. This conjecture was intensively studied and confirmed in many cases. we prove the unstable intersection conjecture in all the remaining cases except the case $dimX=dimY=3$, $dimX\times Y=4$ and $n=5$, which still remains open.