Maps to the projective plane

Abstract

We prove the projective plane [math] is an absolute extensor of a finite-dimensional metrizable space [math] if and only if the cohomological dimension mod [math] of [math] does not exceed [math] . 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 [math] (based at the inclusion) as being isomorphic to either [math] or [math] for [math] . Double surgery and the above fact yield the proof.