Homotopies of maps of suspended real and complex projective spaces and their cohomotopy groups (Q2661700)

For a connected CW complex \(X\) with \(\dim X\leq 2n-2\), let \(\pi^n(X)\) denote the \(n\)-th cohomotopy group given by the set \(\pi^n(X)=[X,S^n]\) of homotopy classes from \(X\) to \(S^n\). Let \(E^k\) denote the \(k\)-fold reduced suspension and we denote by \(\mathbb{F}\mathrm{P}^m\) the \(m\) dimensional real or complex projective space for \(\mathbb{F}=\mathbb{R}\) or \(\mathbb{F}=\mathbb{C}\). For connected spaces \(X\) and \(Y\), let \(M(X,Y)\) denote the space of all continuous maps from \(X\) to \(Y\) with the compact-open topology. In this paper, the authors determine the cohomotopy groups \(\pi^n(E^k\mathbb{F}\mathrm{P}^m)\) for some \(k\geq 0\) and certain \(m,n\geq 1\). As an application they also classify the path components of the spaces \(M(E^k\mathbb{R}\mathrm{P}^m,\mathbb{C}\mathrm{P}^n)\) and \(M(E^k\mathbb{C}\mathrm{P}^m,\mathbb{R}\mathrm{P}^n)\) for \(k\geq 0\) and certain \(m,n\geq 1\). Their method is based on the careful analysis of exact sequences induced from the standard Puppe sequences of \(\mathbb{F}\mathrm{P}^m\).