Certain homotopy properties related to $\text{map}(\Sigma^n \mathbb{C} P^2,S^m)$

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 $\pi^{n+m}(\Sigma^n \mathbb{C} P^2)$ for $m=6,7$. Using these results, we classify path components of the spaces $map(\Sigma^n \mathbb{C} P^2,S^m)$ up to homotopy equivalent. We also determine the generalized Gottlieb groups $G_n(\mathbb{C} P^2,S^m)$. Finally, we compute homotopy groups of mapping spaces $map(\Sigma^n \mathbb{C}P^2,S^m;f)$ for all generators $[f]$ of $[\Sigma^n \mathbb{C} P^2,S^m]$, and Gottlieb groups of mapping components containing constant map $map(\Sigma^n \mathbb{C} P^2,S^m;0)$.