Strong surjections from two-complexes with odd order top-cohomology onto the projective plane (Q6171186)

In [New York J. Math. 27, 615--630 (2021; Zbl 1471.55002)], \textit{M. C. Fenille} and \textit{D. L. Gonçalves} described representatives for all free and based homotopy classes of maps from the model two-complex of the group presentation \(\langle x,y \mid x^{k+1}yxy \rangle\) with \(k \geq 1\) odd into the projective plane. A \textit{strongly surjective map} is a map whose free homotopy class contains only surjective maps. In the paper under review, the authors develop the homotopy set \([K; \mathbb{R}P^2]\) and a classification of the homotopy classes \([f] \in [K; \mathbb{R}P^2]\) according to the property to be a strongly surjective map, where \(K\) is a finite and connected two-dimensional \(CW\)-complex. More precisely, given a finite and connected two-dimensional \(CW\)-complex \(K\) with fundamental group \(\prod\) and second integral cohomology group \(H^2(K; \mathbb{Z})\) finite of odd order, the authors show that for each local integer coefficient system \(\alpha : \prod \rightarrow \mathrm{Aut}(\mathbb{Z}) \cong \mathbb{Z}_2\) over \(K\), the corresponding twisted cohomology group \(H^2 (K; _\alpha{ \mathbb{Z}})\) is finite of odd order \(\mathfrak{c}^*(\alpha)\), and that there exists a natural bijective function from the set \([K; \mathbb{R}P^2]_\alpha^*\) of the based homotopy classes of based maps inducing \(\alpha\) on \(\pi_1\) into \(H^2(K; _\alpha{\mathbb{Z}})\). More interestingly, they also prove that the homotopy set \([K; \mathbb{R}P^2]_\alpha\) of the (free) homotopy classes of based maps inducing \(\alpha\) on \(\pi_1\) is finite of order \(\mathfrak{c}(\alpha) = (\mathfrak{c}^* (\alpha)+ 1)/2\), and that all but one of the homotopy classes \([f] \in [K; \mathbb{R}P^2]_\alpha\) are strongly surjective.