Strong surjections from two-complexes with odd order top-cohomology onto the projective plane

Given a finite and connected two-dimensional CW complex K with fundamental group $$\Pi $$ second integer cohomology $$H^2(K;\mathbb {Z})$$ of odd order, we prove that: (1) for each local coefficient system $$\alpha :\Pi \rightarrow \textrm{Aut}(\mathbb over K, the corresponding twisted $$H^2(K;_{\alpha }\!\mathbb is say order $$\mathfrak {c}^{*}(\alpha )$$ , there exists natural function—which resemble that one defined by degree—from set $$[K;\mathbb {R}\textrm{P}^2]_{\alpha }^{*}$$ based homotopy classes maps inducing on $$\pi _1$$ into which bijection; (2) }$$ (free) {c}(\alpha )=(\mathfrak )+1)/2$$ ; (3) all but $$[f]\in [K;\mathbb are strongly surjective, they characterized non-nullity induced homomorphism $$f^{*}:H^2(\mathbb {R}\textrm{P}^2;_{\varrho {Z})\rightarrow H^2(K;_{\alpha where $$\varrho nontrivial projective plane. Also some calculations provided several two-complexes actions allowing to compare .