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

Abstract

Given a finite and connected two-dimensional $CW$-complex $K$ with fundamental group $\Pi$ and second integer cohomology group $H^2(K;\mathbb{Z})$ finite of odd order, we prove that: (1) for each local integer coefficient system $\alpha:\Pi\to{\rm Aut}(\mathbb{Z})$ over $K$, the corresponding twisted cohomology group $H^2(K;_{\alpha}\!\mathbb{Z})$ is finite of odd order, we say order $\mathbb{C}^{\ast}(\alpha)$, and there exists a natural function -- which resemble that one defined by the twisted degree -- from the set $[K;\mathbb{R}P^2]_{\alpha}^{\ast}$ of the based homotopy classes of based maps inducing $\alpha$ on $\pi_1$ into $H^2(K;_{\alpha}\!\mathbb{Z})$, which is a bijection; (2) the set $[K;\mathbb{R}P^2]_{\alpha}$ of the (free) homotopy classes of based maps inducing $\alpha$ on $\pi_1$ is finite of order $\mathbb{C}(\alpha)=(\mathbb{C}^{\ast}(\alpha)+1)/2$; (3) all but one of the homotopy classes $[f]\in[K;\mathbb{R}P^2]_{\alpha}$ are strongly surjective, and they are characterized by the non-nullity of the induced homomorphism $f^{\ast}:H^2(\mathbb{R}P^2;_{\varrho}\!\mathbb{Z})\to H^2(K;_{\alpha}\!\mathbb{Z})$, where $\varrho$ is the nontrivial local integer coefficient system over the projective plane. Also some calculations of the groups $H^2(K;_{\alpha}\!\mathbb{Z})$ are provided for several two-complexes $K$ and actions $\alpha$, allowing to compare $H^2(K;\mathbb{Z})$ and $H^2(K;_{\alpha}\!\mathbb{Z})$ for nontrivial $\alpha$.