Basic Fibrations

Consider now the natural projection proj _k,r : $$\mathbb{F}_k (M) \to \mathbb{F}_r (M)$$, r < k that sends (x₁, …, x _k ) to (x₁, …, 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 = {q₁, …, q _r } and q = (q₁, …, q _r ), is the basepoint of $$\mathbb{F}_r (M)$$.