Maps to the projective plane

Abstract

We prove the projective plane $ℝP^2$ is an absolute extensor of a finite-dimensional metrizable space $X$ if and only if the cohomological dimension mod $2$ of $X$ does not exceed $1$. This solves one of the remaining difficult problems (posed by A N Dranishnikov) in Extension Theory. One of the main tools is the computation of the fundamental group of the function space $Map\left(ℝP^n,ℝP^{n+1}\right)$ (based at the inclusion) as being isomorphic to either ${\mathbb{Z}}_4$ or ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$ for $n\ge 1$. Double surgery and the above fact yield the proof.