Evaluation fibrations and path-components of the map space $M(\mathbb S^n+k,\mathbb S^n)$ for $0\le k\le 7$

M.Golasi'nski

Let $M(\mathbb{S}^m,\mathbb{S}^n)$ be the space of maps of the $m$-sphere $\mathbb{S}^m$ into the $n$-sphere $\mathbb{S}^n$ with $m,n\ge 1$. We estimate the number of homotopy types of path-components of $M(\mathbb{S}^{n+k},\mathbb{S}^n)$ and fibre homotopy types of evaluation fibrations $\omega_\alpha\colon M_\alpha(\mathbb{S}^{n+k},\mathbb{S}^n)\to \mathbb{S}^n$ for $\alpha\in\pi_{n+k}(\mathbb{S}^n)$ with $0\le~k\le~7.$ Further, the number of strongly homotopy types of $\omega_\alpha\colon M_\alpha(\mathbb{S}^{n+k}, \mathbb{S}^n)\to \mathbb{S}^n$ for $0\le~k\le~7$ is determined.

1. M. Golaśiński and J. Mukai, Gottlieb groups of spheres, Topology 47 (2008), 399-430.

2. D. Gottlieb, A certain subgroup of the fundamental group, Amer. J. of Math.\ 87 (1965), 840-856.

3. D. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. of Math. 91 (1969), 729-756.

4. V.L. Hansen, Equivalence of evaluation fibrations, Invent. Math. 23 (1974), 163-171.

5. V.L. Hansen, The homotopy problem for the components in the space of maps of the $n$-sphere, Quart. J. Math. Oxford Ser. (2) 25 (1974), 313-321.

6. S.T. Hu, Homotopy theory, Academic Press, 1959.

7. G. Lupton and S.B. Smith, Criteria for components of a function space to be homtopy equivalence, preprint.

8. H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies 49, Princeton, 1962.

	Evaluation fibrations and path-components of the map space $M(\mathbb S^n+k,\mathbb S^n)$ for $0\le k\le 7$
Author	M.Golasi'nski marek@mat.uni.torun.pl, marek@matman.uwm.edu.pl Faculty of Mathematics and Computer Science,Nicolaus Copernicus University,Chopina 12/18, 87-100 Toruń, Poland, , Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Żołnierska 14, 10-561 Olsztyn, Poland
Abstract	Let $M(\mathbb{S}^m,\mathbb{S}^n)$ be the space of maps of the $m$-sphere $\mathbb{S}^m$ into the $n$-sphere $\mathbb{S}^n$ with $m,n\ge 1$. We estimate the number of homotopy types of path-components of $M(\mathbb{S}^{n+k},\mathbb{S}^n)$ and fibre homotopy types of evaluation fibrations $\omega_\alpha\colon M_\alpha(\mathbb{S}^{n+k},\mathbb{S}^n)\to \mathbb{S}^n$ for $\alpha\in\pi_{n+k}(\mathbb{S}^n)$ with $0\le~k\le~7.$ Further, the number of strongly homotopy types of $\omega_\alpha\colon M_\alpha(\mathbb{S}^{n+k}, \mathbb{S}^n)\to \mathbb{S}^n$ for $0\le~k\le~7$ is determined.
Keywords	evaluation fibration, path-component, map space
DOI	doi:10.30970/ms.31.2.189-194
Reference	1. M. Golaśiński and J. Mukai, Gottlieb groups of spheres, Topology 47 (2008), 399-430. 2. D. Gottlieb, A certain subgroup of the fundamental group, Amer. J. of Math.\ 87 (1965), 840-856. 3. D. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. of Math. 91 (1969), 729-756. 4. V.L. Hansen, Equivalence of evaluation fibrations, Invent. Math. 23 (1974), 163-171. 5. V.L. Hansen, The homotopy problem for the components in the space of maps of the $n$-sphere, Quart. J. Math. Oxford Ser. (2) 25 (1974), 313-321. 6. S.T. Hu, Homotopy theory, Academic Press, 1959. 7. G. Lupton and S.B. Smith, Criteria for components of a function space to be homtopy equivalence, preprint. 8. H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies 49, Princeton, 1962.
Pages	189-194
Volume	31
Issue	2
Year	2009
Journal	Matematychni Studii
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