On 𝑅^{∞}(𝑄^{∞})-manifold bundles over CW complexes

Abstract

Let Λ ∈ C W ( C ) ∪ C W ( M ) \Lambda \in \mathcal {C}\mathcal {W}(\mathcal {C}) \cup \mathcal {C}\mathcal {W}(\mathcal {M}) be a pseudo CW complex generated either by Hausdorff compact spaces or by metric spaces. In the theory of manifolds modeled on R ∞ {R^\infty } or Q ∞ {Q^\infty } , we will prove the Λ \Lambda -fiber-preserving versions of the following: Equivalences among the notions of D D -sets, D ∗ {D^{\ast }} -sets and infinite deficient sets; relative stability theorems; relative deformation of homotopy equivalences to homeomorphisms; strong unknotting theorem for D D -embeddings; and α \alpha -approximation theorems.