On the smoothing theory delooping of disc diffeomorphism and embedding spaces

Abstract The celebrated Morlet–Burghelea–Lashof–Kirby–Siebenmann smoothing theory theorem states that the group $$\textrm{Diff}_\partial (D^n)$$ Diff ∂ ( D n ) of diffeomorphisms of a disc $$D^n$$ D n relative to the boundary is equivalent to $$\Omega ^{n+1}\left( \textrm{PL}_n/\textrm{O}_n\right) $$ Ω n + 1 PL n / O n for any $$n\ge 1$$ n ≥ 1 and to $$\Omega ^{n+1}\left( \textrm{TOP}_n/\textrm{O}_n\right) $$ Ω n + 1 TOP n / O n for $$n\ne 4$$ n ≠ 4 . We revise smoothing theory results to show that the delooping generalizes to different versions of disc smooth embedding spaces relative to the boundary, namely the usual embeddings, those modulo immersions, and framed embeddings. The latter spaces deloop as $$\textrm{Emb}_\partial ^{fr}(D^m,D^n)\simeq \Omega ^{m+1}\left( \textrm{O}_n\backslash \!\!\backslash \textrm{PL}_n/\textrm{PL}_{n,m}\right) \simeq \Omega ^{m+1}\left( \textrm{O}_n\backslash \!\!\backslash \textrm{TOP}_n/\textrm{TOP}_{n,m}\right) $$ Emb ∂ fr ( D m , D n ) ≃ Ω m + 1 O n \ \ PL n / PL n , m ≃ Ω m + 1 O n \ \