Strongly surjective maps from certain two-complexes with trivial top-cohomology onto the projective plane (Q2662945)

The Hopf-Whitney Classification Theorem asserts that the set of homotopy classes of maps from an \(n\)-dimensional CW-complex \(X\) into an \((n-1)\)-connected \(n\)-simple space \(Y\) is in one-to-one correspondence with the cohomology group \(H^n (X; \pi_n (Y))\). In particular, there exists a map from a connected \(n\)-dimensional CW-complex \(X\) onto the \(n\)-sphere \(S^n\) whose free homotopy class contains only surjective maps if and only if \(H^n(X;\mathbb{Z}) \neq 0\). Such a map is said to be a \textit{strong surjection}. In [Fundam. Math. 192, No. 3, 195--214 (2006; Zbl 1111.55001); Cent. Eur. J. Math. 6, No. 4, 497--503 (2008; Zbl 1153.55002)], \textit{C. Aniz} investigated the strong connectivity of maps of some \(3\)-dimensional complex into \(3\)-dimensional manifolds such as the product \(S^1 \times S^2\), the \(S^1\)-bundle over \(S^2\) and the orbit space of \(S^3\) with respect to the action of the quaternion group. In [Topology Appl. 210, 63--69 (2016; Zbl 1351.55003)], \textit{M. C. Fenille} constructed a countable collection of connected \(2\)-dimensional finite CW-complexes with trivial second integral cohomology group, and a map from each of them onto the torus whose homotopy class contains only strong surjections. Let \(K\) be a connected \(2\)-dimensional finite CW-complex of the group presentation \(\mathcal{P} = \langle x,y \mid x^{k+1}yxy \rangle\) with \(k \geq 1\) odd, and let \(\mathbb{R}P^2\) be the real projective space in dimension \(2\). Let \([K, \mathbb{R}P^2]\) (resp. \([K, \mathbb{R}P^2]^*\)) be the set of free (resp. based) homotopy classes of maps (resp. based maps) \(K \rightarrow \mathbb{R}P^2\). In this paper under review, the authors show that (1) if \(k=1\), then \([K, \mathbb{R}P^2] \approx [K, \mathbb{R}P^2]^* \approx \{1\} \sqcup \{\bar 0\}\) and there does not exist a strong surjection from \(K\) onto the real projective space \(\mathbb{R}P^2\); and (2) if \(k = 2p-1 \geq 3\), then \([K, \mathbb{R}P^2]^* \approx \{1\} \sqcup \mathbb{Z}_k\) and \([K, \mathbb{R}P^2] \approx \{1\} \sqcup \mathbb{Z}_p\), and the free homotopy class corresponding to \(1\) contains a non-surjective map and the remaining \(p-1\) classes contain only strong surjections. Here, \(1\) and \(0\) is originated from Hom\((\pi_1 (K), \mathbb{Z}_2 ) = \{1, \beta\}\) in which \(1\) is the trivial homomorphism and \(\beta : \pi_1 (K) \rightarrow \mathrm{Aut}(\mathbb{Z})\) is the homomorphism given by \(\beta(\bar x) = 1\) and \(\beta(\bar y) = -1\), where \(\bar x\) and \(\bar y\) are two generators of \(\pi_1 (K) = F(x,y)/N(x^{k+1}yxy)\) presented by \(\mathcal{P}\).