On the smoothing theory delooping of disc diffeomorphism and embedding spaces

The celebrated Morlet-Burghelea-Lashof-Kirby-Siebenmann smoothing theory theorem states that the group $\mathrm{Diff}\partial(D^n)$ of diffeomorphisms of a disc $D^n$ relative to the boundary is equivalent to $Ω^{n+1}\left(\mathrm{PL}n/\mathrm{O}n\right)$ for any $n\geq 1$ and to $Ω^{n+1}\left(\mathrm{TOP}n/\mathrm{O}n\right)$ for $n\neq 4$. 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 $\mathrm{Emb}\partial^{fr}(D^m,D^n)\simeqΩ^{m+1}\left(\mathrm{O}n\backslash!!\backslash\mathrm{PL}n/\mathrm{PL}{n,m}\right)\simeq Ω^{m+1}\left(\mathrm{O}n\backslash!!\backslash\mathrm{TOP}n/\mathrm{TOP}{n,m}\right)$ for any $n\geq m\geq 1$ ($n\neq 4$ for the second equivalence), where the left-hand side in the case $n-m=2$ or $(n,m)=(4,3)$ should be replaced by the union of the path-components of $\mathrm{PL}$-trivial knots (framing being disregarded). Moreover, we show that for $n\neq 4$, the delooping is compatible with the Budney $E{m+1}$-action. We use this delooping to combine the Hatcher $\mathrm{O}{m+1}$-action and the Budney $E{m+1}$-action into a framed little discs operad $E{m+1}^{\mathrm{O}{m+1}}$-action on $\mathrm{Emb}\partial^{fr}(D^m,D^n)$.

💡 Research Summary

The paper revisits the classical smoothing theory result—often attributed to Morlet, Burghelea, Lashof, Kirby, and Siebenmann—that the group of diffeomorphisms of a disc fixing the boundary, Diff₍∂₎(Dⁿ), is equivalent to the (n + 1)-fold loop space of the homotopy quotient PLₙ/Oₙ (and, for n ≠ 4, also to Ωⁿ⁺¹(TOPₙ/Oₙ). The authors extend this “delooping” phenomenon from the diffeomorphism group to several natural spaces of smooth embeddings of a disc Dᵐ into a larger disc Dⁿ, all relative to the boundary. The three embedding variants considered are: (i) ordinary smooth embeddings Emb₍∂₎(Dᵐ,Dⁿ), (ii) embeddings modulo immersions (the homotopy fibre of the immersion map), and (iii) framed embeddings Emb₍∂₎^{fr}(Dᵐ,Dⁿ), where a framing of the normal bundle is prescribed.

The central technical achievement is a family of equivalences that express these embedding spaces as iterated loop spaces of double coset constructions involving the orthogonal group Oₙ and the PL or TOP versions of the Stiefel manifolds Vₙ,ₘ = Oₙ/Oₙ₋ₘ. Concretely, for any 1 ≤ m ≤ n with n ≠ 4, \