A function à la Hopf-Whitney that detects or not strong surjectivity

Abstract

Given a finite and connected two-dimensional complex $K$ and a homomorphism $\beta\in\hom(\pi_1(K);\mathbb Z_2)$, we consider the function $\Phi_{\beta}\colon [K;\RP^2]^{\ast}_{\beta}\to H^2(K;{}_{\beta}\mathbb Z)$ defined by $[f]^{\ast}\mapsto f^{\ast}(\nu)$, where $\nu$ is a preferred generator of the twisted cohomology group $H^2(\mathbb{R}\mathrm{P}^2;{}_{\varrho}\Z)$. We prove that $\Phi_{\beta}$ is a $\kappa_{\beta}$-to-one function, where $\kappa_{\beta}$ is the order of the kernel of the multiplication by $2$ on $H^2(K;{}_{\beta}\mathbb Z)$, and we present necessary and sufficient conditions for both: $\Phi_{\beta}$ to be injective and $\Phi_{\beta}$ to be surjective. Furthermore, we prove that $\Phi_{\beta}$ detects strong surjectivity if and only if either $0\notin{\rm im}(\Phi_{\beta})$ or the set $[K;\mathbb{R}\mathrm{P}^2]^{\ast}_{\beta}$ contains $\kappa_{\beta}$ classes having a representative given by a composite $K\to S^1\hookrightarrow\mathbb{R}\mathrm{P}^2$.