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

TL;DR

This paper computes cohomotopy groups and classifies homotopy types of mapping spaces from suspended complex projective planes to spheres, also determining Gottlieb groups and homotopy groups of these mapping spaces.

Contribution

It provides explicit calculations of cohomotopy groups, classifies path components of mapping spaces, and determines Gottlieb groups for maps from suspended complex projective planes to spheres.

Findings

Cohomotopy groups of $ ext{Suspended } ext{CP}^2$ computed for specific degrees.

Classification of mapping space components up to homotopy.

Explicit homotopy groups and Gottlieb groups of mapping spaces obtained.

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 $map(\Sigma^n \mathbb{C} P^2,S^m;0)$.