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Basic Fibrations

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Abstract

Consider now the natural projection proj k,r : \(\mathbb{F}_k (M) \to \mathbb{F}_r (M)\), r < k that sends (x 1, …, x k ) to (x 1, …, x r ), where M is a connected manifold of dimension m. With the configuration spaces being regarded as the space of imbeddings of the sets k and r, respectively, one sees that these projections are just the restriction maps induced by the injection r = {1, …, r} ⊂ k = {1, …, k}. Hence, according to [107, Thom] (see also [87, Palais]), these projections are locally trivial fibrations. As the fibration of \(\mathbb{F}_k (M)\) over \(\mathbb{F}_r (M)\) plays a central role in the study of the geometry and topology of these configuration spaces ([43, Fadell-Neuwirth], [17, Cohen]), a simple and direct proof of the fact that proj k,r : \(\mathbb{F}_k (M) \to \mathbb{F}_r (M)\) is locally trivial, independent of [107, Thom] (or [87, Palais]), is in order. In §1 below, we give such a proof. The structure group will be a subgroup of Top(M), and its fiber the configuration space \(\mathbb{F}_{k - r} (M - Q_r )\), where Q r = {q 1, …, q r } and q = (q 1, …, q r ), is the basepoint of \(\mathbb{F}_r (M)\).

Keywords

  • Homogeneous Space
  • Tangent Bundle
  • Configuration Space
  • Natural Projection
  • Principal Bundle

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2001 Springer-Verlag Berlin Heidelberg

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Fadell, E.R., Husseini, S.Y. (2001). Basic Fibrations. In: Geometry and Topology of Configuration Spaces. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56446-8_2

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  • DOI: https://doi.org/10.1007/978-3-642-56446-8_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63077-4

  • Online ISBN: 978-3-642-56446-8

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