2017
Том 69
№ 6

All Issues

Evaluation Fibrations and Path-Components of the Mapping Space $M\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)$ for $8 ≤ k ≤ 13$

de Melo Thiago, Golasinski M.

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Abstract

Let $M\left( {{{\mathbb{S}}^{m}},{{\mathbb{S}}^n}} \right)$ be the space of maps from the $m$-sphere ${\mathbb{S}}^{m}$ into the $n$-sphere ${\mathbb{S}}^{n}$ with $m,n ≥ 1$. We estimate the number of homotopy types of path-components $M_{\alpha}\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)$ and fiber homotopy types of the evaluation fibrations ${\omega_{\alpha }}:{M_{\alpha }}\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)\to {{\mathbb{S}}^n}$ for $8 ≤ k ≤ 13$ and $\alpha \in {\pi_{n+k }}\left( {{{\mathbb{S}}^n}} \right)$ extending the results of [Mat. Stud. - 2009. - 31, № 2. -P. 189-194]. Further, the number of strong homotopy types of ${\omega_{\alpha }}:{M_{\alpha }}\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)\to {{\mathbb{S}}^n}$ for $8 ≤ k ≤ 13$ is determined and some improvements of the results from [Mat. Stud. - 2009. - 31, № 2. - P. 189-194] are obtained.

English version (Springer): Ukrainian Mathematical Journal 65 (2013), no. 8, pp 1141-1154.

Citation Example: de Melo Thiago, Golasinski M. Evaluation Fibrations and Path-Components of the Mapping Space $M\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)$ for $8 ≤ k ≤ 13$ // Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1023-1034.

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