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. (Ukrains’kyi Matematychnyi Zhurnal, 2013)
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 $M_{α}\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)$ and fiber homotopy types of the evaluation fibrations ${ω_{α }}:{M_{α }}\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)→ {{\mathbb{S}}^n}$ for $8 ≤ k ≤ 13$ and $α \in {π_{n+k }}\left( {{{\mathbb{S}}^n}} \right)$ extending the results of [Mat. Stud. - 2009. - 31, № 2. -P. 189-194]. Further, the number of strong homotopy types of ${ω_{α }}:{M_{α }}\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)→ {{\mathbb{S}}^n}$ for $8 ≤ k ≤ 13$ is determined and some improvements of the results from [Mat. Stud. - 2009. - 31, № 2. - P. 189-194] are obtained.