# 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## Preview

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