We prove the projective plane is an absolute extensor of a finite-dimensional metrizable space if and only if the cohomological dimension mod of does not exceed . 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 (based at the inclusion) as being isomorphic to either or for . Double surgery and the above fact yield the proof.
"Maps to the projective plane." Algebr. Geom. Topol. 9 (1) 549 - 568, 2009. https://doi.org/10.2140/agt.2009.9.549