Convenient maps from one-relator model two-complexes into the real projective plane

Abstract

Let $f$ be a map from a one-relator model two-complex $K_{\mathcal{P}}$ into the real projective plane. The composition $\varrho\circ f_{\#}$ of the homomorphism $f_{\#}$ induced by $f$ on fundamental groups with the action $\varrho$ of $\pi_1(\mathbb{R}\mathrm{P}^2)$ over $\pi_2(\mathbb{R}\mathrm{P}^2)$ provides a local integer coefficient system $f_{\#}^{\varrho}$ over $K_{\mathcal{P}}$. We prove that if the twisted integer cohomology group $H^2(K_{\mathcal{P}};_{f_{\#}^{\varrho}}\!\mathbb Z)=0$, then $f$ is homotopic to a non-surjective map. As an intermediary step for the proof, we show that if $H^2(K_{\mathcal{P}};_{\beta}\!\mathbb Z)=0$ for some local integer coefficient system $\beta$ over $K_{\mathcal{P}}$, then $K_{\mathcal{P}}$ is aspherical.