Certain homotopy properties related to $\text{map}(Σ^n \mathbb{C} P^2,S^m)$ (1501.03242v2)
Abstract: For given spaces $X$ and $Y$, let $map(X,Y)$ and $map_\ast(X,Y)$ be the unbased and based mapping spaces from $X$ to $Y$, equipped with compact-open topology respectively. Then let $map(X,Y;f)$ and $map_\ast(X,Y;g)$ be the path component of $map(X,Y)$ containing $f$ and $map_\ast(X,Y)$ containing $g$, respectively. In this paper, we compute cohomotopy groups of suspended complex plane $\pi{n+m}(\Sigman \mathbb{C} P2)$ for $m=6,7$. Using these results, we classify path components of the spaces $map(\Sigman \mathbb{C} P2,Sm)$ up to homotopy equivalent. We also determine the generalized Gottlieb groups $G_n(\mathbb{C} P2,Sm)$. Finally, we compute homotopy groups of mapping spaces $map(\Sigman \mathbb{C}P2,Sm;f)$ for all generators $[f]$ of $[\Sigman \mathbb{C} P2,Sm]$, and Gottlieb groups of mapping components containing constant map $map(\Sigman \mathbb{C} P2,Sm;0)$.