The trivial homotopy class of maps from two-complexes into the real projective plane
\textit{C. Aniz} [Fundam. Math. 192, No. 3, 195--214 (2006; Zbl 1111.55001)] studied the root free problem for a map \(f : K^m \rightarrow M^m\) from a complex \(K^m\) to a manifold \(M^m\), where \(K^m\) and \(M^m\) have the same dimension \(m\), which is greater than or equal to \(3\), and showed in [Cent. Eur. J. Math. 6, No. 4, 497--503 (2008; Zbl 1153.55002)] that if \(X\) is a \(CW\)-complex of dimension \(3\) such that \(H^3 (X;\mathbb{Z})=0\) and if \(M_{Q_8}\) is the orbit space of the 3-sphere \(S^3\) with respect to the action of the quaternion group \(Q_8\) determined by the inclusion \(Q_8\subseteq S^3\), then there is no map \(f : X \rightarrow M_{Q_8}\) which is a strong surjection. In this paper, by using the Hopf-Whitney Classification Theorem and a specific problem in Topological Root Theory, the author shows that the following statements are equivalent for a finite and connected two-dimensional \(CW\)-complex \(K\) with trivial second integer cohomology group: (1) Every based map \(f : K \rightarrow \mathbb{R}P^2\) from \(K\) into the real projective plane \(\mathbb{R}P^2\) is based homotopic to a constant map (i.e., inessential); (2) The skeleton pair \((K, K^1)\) is homotopy equivalent to that of a model two-complex induced by a balanced group presentation; (3) The number of \(2\)-cells of \(K\) is equal to the first Betti number of its one-skeleton \(K^1\); (4) \(K\) is acyclic; (5) Every based map from \(K\) into the circle \(S^1\) is inessential.
scientific article; zbMATH DE number 6700564