Homotopy properties of $\text{map}(\Sigma^n \mathbb C P^2,S^m)$
J. Korean Math. Soc. 2021 Vol. 58, No. 3, 761-790
Published online January 19, 2021
Printed May 1, 2021
Jin-ho Lee
Data Science, Pulse9. Inc. 449
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 $\pi^{n+m}(\Sigma^n \C P^2)$ for $m=6,7$. Using these results, we classify path components of the spaces $map(\Sigma^n \C P^2,S^m)$ up to homotopy equivalence. We also determine the generalized Gottlieb groups $G_n(\C P^2,S^m)$. Finally, we compute homotopy groups of mapping spaces $map(\Sigma^n \mathbb{C}P^2,S^m;f)$ for all generators $[f]$ of $[\Sigma^n \C P^2,S^m]$, and Gottlieb groups of mapping components containing constant map $map(\Sigma^n \C P^2,S^m;\ast)$.
Keywords : Composition methods, homotopy groups of mapping spaces, cohomotopy groups, Gottlieb groups, evaluation fibrations
MSC numbers : Primary 55Q55; Secondary 55P15
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