  We prove the projective plane $\rp^2$ is an absolute extensor of a
finite-dimensional metric 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
$\Map(\rp^n,\rp^{n+1})$ (based at inclusion) as being isomorphic to either
$\Z_4$ or $\Z_2\oplus\Z_2$ for $n\ge 1$. Double surgery and the above fact
yield the proof.
