Evaluation fibrations and path-components of the mapping space \(M(\mathbb{S}^{n+k},\mathbb{S}^n)\) for \(8\leq k\leq 13\)
Let \(M(\mathbb{S}^{n+k},\mathbb{S}^n)\) denote the space of all continuous maps from the \((n+k)\)-sphere \(\mathbb{S}^{n+k}\) into the \(n\)-sphere \(\mathbb{S}^n\) with the compact open topology. For an element \(\alpha\in \pi_{n+k}(\mathbb{S}^n)\), let \(M_{\alpha}(\mathbb{S}^{n+k},\mathbb{S}^n)\) be its path-component containing \(\alpha\). In this paper, the authors consider the problem to estimate the number of homotopy types of path-components \(M_{\alpha}(\mathbb{S}^{n+k},\mathbb{S}^n)\) and their corresponding fibre homotopy types. In particular, they estimate the number of homotopy types of \(M_{\alpha}(\mathbb{S}^{n+k},\mathbb{S}^n)\) explicitly for \(8\leq k\leq 13\). Moreover, they also improve the number of strong homotopy types of the evaluation map \(\omega_{\alpha}:M_{\alpha}(\mathbb{S}^{n+k\mathbb{S}^n})\to \mathbb{S}^n\) for \(8\leq k\leq 13\).
scientific article; zbMATH DE number 6508259
scientific article