Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2021; 58(3): 761-790

Published online May 1, 2021 https://doi.org/10.4134/JKMS.j200249

Copyright © The Korean Mathematical Society.

Homotopy properties of $\text{map}(\Sigma^n \mathbb C P^2,S^m)$

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