Maps to the projective plane

We prove the projective plane RP² 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(RPⁿ, RPⁿ⁺¹) (based at the inclusion) as being isomorphic to either Z₄ or Z₂ Z₂ for n ≥ 1. Double surgery and the above fact yield the proof.