On the smoothing theory delooping of disc diffeomorphism and embedding spaces

TL;DR

The work extends Morlet–Burghelea–Lashof–Kirby–Siebenmann smoothing theory by proving delooping results for disc diffeomorphism and embedding spaces relative to the boundary, and by embedding these deloopings into framed Little Discs operad actions. It shows that Embeddings and framed embeddings deloop as iterated loop spaces of quotients like $Ω^{m+1}(O_n\backslash\!\!\!/TOP_n/ TOP_{n,m})$, with TOP, PL, and PD variants treated and special care given to the four-dimensional case. The authors establish compatibilities with Budney’s $E_{m+1}$-action and produce a framed little discs operad action on $Emb_\partial^{fr}(D^m,D^n)$, unifying smoothing theory with operadic/delooping frameworks and connecting diffeomorphism groups to topological embedding calculus via loop-space models. The results illuminate the structure of embedding spaces across codimensions, including invertible components in critical cases ($n-m=2$) and the distinctive behavior in dimension four, with implications for understanding smooth, PL, and topological knot theories through a unified homotopy-theoretic lens.

Abstract

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 $\mathrm{Emb}_\partial^{fr}(D^m,D^n)$.

Theorems & Definitions (54)

• Theorem A
• Theorem B
• Theorem C
• Corollary 1.1
• Remark 1.2
• Theorem 2.1
• proof
• Corollary 2.2
• Theorem 2.3
• Corollary 2.4