Abstract
Under certain conditions, we describe the homotopy type of the homotopy fibre of the inclusion map \(F_n(X)\hookrightarrow \prod _1^nX\) for the nth configuration space \(F_n(X)\) of a topological manifold X without boundary such that \(\mathrm{{dim}}(X)\ge 3\). We then apply our results to the cases where either the universal covering of X is contractible or X is an orbit space \(\mathbb {S}^k/G\) of a tame, free action of a Lie group G on the k-sphere \(\mathbb {S}^k\). If the group G is finite and k is odd, we give a full description of the long exact sequence in homotopy of the homotopy fibration of the inclusion map \(F_n(\mathbb {S}^k/G)\hookrightarrow \prod _1^n\mathbb {S}^k/G\).
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Acknowledgments
The authors would like to thank the referee for having carefully read the paper and for his or her suggestions. The first and second authors are deeply indebted to the Institute for Mathematical Sciences, National University of Singapore, for support to attend the Combinatorial and Toric Homotopy Conference (Singapore, 23–29 August 2015) in honour of Frederick Cohen, where many of the ideas for this paper originated. Further work on the paper took place during the visit of the second author to the Institute of Mathematics, Kazimierz Wielki University, Bydgoszcz (Poland) during the period 31 August–12 September 2015, and during the visit of the third author to the Departamento de Matemática do IME—Universidade de São Paulo (Brasil) during the period 19 October–3 November 2015. These two visits were partially supported by FAPESP-Fundação de Amparo a Pesquisa do Estado de São Paulo, Projeto Temático Topologia Algébrica, Geométrica e Diferencial 2012/24454-8 (Brasil), and by the CNRS/FAPESP programme no 226555 (France) and no 2014/50131-7 (Brasil), respectively.
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This paper is dedicated to Professor S. Gitler, for his outstanding work in homotopy theory, his readiness to encourage the scientific progress of colleagues and of institutions, his unwavering intellectual honesty, and his warm friendship.
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Golasiński, M., Lima Gonçalves, D. & Guaschi, J. On the homotopy fibre of the inclusion map \(F_n(X)\hookrightarrow \prod _1^nX\) for some orbit spaces X . Bol. Soc. Mat. Mex. 23, 457–485 (2017). https://doi.org/10.1007/s40590-016-0150-6
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DOI: https://doi.org/10.1007/s40590-016-0150-6
Keywords
- Configuration space
- Free action of a group
- Homotopy fibre
- Homotopy pullback
- Lie group
- Manifold
- Sphere
- Orbit space
- Whisker map