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Basic Fibrations

  • Edward R. Fadell
  • Sufian Y. Husseini
Part of the Springer Monographs in Mathematics book series (SMM)

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Edward R. Fadell
    • 1
  • Sufian Y. Husseini
    • 1
  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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