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

TL;DR

This paper investigates the properties of maps from certain two-dimensional complexes to the projective plane, focusing on their surjectivity and the structure of associated twisted cohomology groups, revealing new relationships between homotopy classes and cohomology.

Contribution

It establishes a bijection between homotopy classes of maps inducing a given action on fundamental groups and twisted cohomology groups, and characterizes strongly surjective maps in this context.

Findings

Most homotopy classes are strongly surjective.

The set of homotopy classes is finite and explicitly computed.

Twisted cohomology groups are finite of odd order and related to map classifications.

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 $\alpha$.