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