Homotopy properties of \(\text{map}(\Sigma^n \mathbb{C}P^2,S^m)\). (English) Zbl 1471.55017
Given spaces \(X\) and \(Y\), let \(\operatorname{map}(X,Y)\) and \(\operatorname{map}_\ast(X,Y)\) be the unbased and based mapping spaces from \(X\) to \(Y\), equipped with compact-open topology, respectively.
H. Kachi et al. [Math. J. Okayama Univ. 43, 105–121 (2001; Zbl 1039.55010)] described cohomotopy groups \(\pi^{m+n}(\Sigma^n\mathbb{C}P^2)\) in the range of \(-5\le m\le 1\) of the \(n\)-suspended complex plane \(\Sigma^n\mathbb{C}P^2\).
The author computes \(\pi^{m+n}(\Sigma^n\mathbb{C}P^2)\) for \(m=6,7\). Using these results, path components \(\operatorname{map}(\Sigma^n\mathbb{C}P^2, \mathbb{S}^m; f)\) of the mapping space \(\operatorname{map}(\Sigma^n\mathbb{C}P^2, \mathbb{S}^m)\) (for \(f\in \operatorname{map}(\Sigma^n\mathbb{C}P^2,\mathbb{S}^m)\)) are classified up to homotopy equivalence and the generalized Gottlieb groups \(G_n(\mathbb{C}P^2,\mathbb{S}^m)\) for some low dimensions are determined. Finally, some homotopy groups of the path components \(\operatorname{map}(\Sigma^n\mathbb{C}P^2, \mathbb{S}^m; f)\) for all generators \([f]\) of \(\pi^m(\Sigma^n\mathbb{C}P^2)\) are computed.
H. Kachi et al. [Math. J. Okayama Univ. 43, 105–121 (2001; Zbl 1039.55010)] described cohomotopy groups \(\pi^{m+n}(\Sigma^n\mathbb{C}P^2)\) in the range of \(-5\le m\le 1\) of the \(n\)-suspended complex plane \(\Sigma^n\mathbb{C}P^2\).
The author computes \(\pi^{m+n}(\Sigma^n\mathbb{C}P^2)\) for \(m=6,7\). Using these results, path components \(\operatorname{map}(\Sigma^n\mathbb{C}P^2, \mathbb{S}^m; f)\) of the mapping space \(\operatorname{map}(\Sigma^n\mathbb{C}P^2, \mathbb{S}^m)\) (for \(f\in \operatorname{map}(\Sigma^n\mathbb{C}P^2,\mathbb{S}^m)\)) are classified up to homotopy equivalence and the generalized Gottlieb groups \(G_n(\mathbb{C}P^2,\mathbb{S}^m)\) for some low dimensions are determined. Finally, some homotopy groups of the path components \(\operatorname{map}(\Sigma^n\mathbb{C}P^2, \mathbb{S}^m; f)\) for all generators \([f]\) of \(\pi^m(\Sigma^n\mathbb{C}P^2)\) are computed.
Reviewer: Marek Golasiński (Olsztyn)
Keywords:
composition methods; homotopy groups of mapping spaces; cohomotopy groups; Gottlieb groups; evaluation fibrationsCitations:
Zbl 1039.55010References:
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