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

• Published in: arXiv.org 2021-10
• Main Authors: Fenille, Marcio C, Gonçalves, Daciberg L, Neto, Oziride M
• Format: Article
• Language: English
• Subjects: Homology; Homomorphisms; Integers
• ISSN: 2331-8422

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 \(\alpha\).