A function à la Hopf-Whitney that detects or not strong surjectivity - Fenille, Marcio Colombo; Gonçalves, Daciberg Lima (Topological Methods in Nonlinear Analysis, 2026)
Given a finite and connected two-dimensional complex K and a homomorphism β(π_1(K);Z_2), we consider the function Φ_β[K;^2]^*_β→ H^2(K;_βZ) defined by [f]^*f^*( u), where u is a preferred generator of the twisted cohomology group H^2(RP^2;_ρ). We prove that Φ_β is a κ_β-to-one function, where κ_β is the order of the kernel of the multiplication by 2 on H^2(K;_βZ), and we present necessary and sufficient conditions for both: Φ_β to be injective and Φ_β to be surjective. Furthermore, we prove that Φ_β detects strong surjectivity if and only if either 0 otinim(Φ_β) or the set [K;RP^2]^*_β contains κ_β classes having a representative given by a composite K→ S^1RP^2.