Twisted and Absolute Degrees of Maps into the Projective Plane

For each based map f: X → ℝP² from a closed surface into the real projective plane, we compute its absolute and twisted degrees and describe the action of the fundamental group of ℝP² over the based homotopy class of f. We emphasize the finding that for any nonorientable closed surface, there exists only one based homotopy class of maps from it into ℝP² whose maps have twisted degree zero and absolute degree nonzero–which shows that, unlike the absolute degree, the twisted degree is not able to detect the strong surjectivity in this setting. In all the other scenarios, the absolute degree of each map is either equal to the twisted degree or its absolute value–and so the twisted degree detects strong surjectivity.