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 $\mathrm{Diff}_\partial(D^n)$ of diffeomorphisms of a disc $D^n$ relative to the boundary is equivalent to $\Omega^{n+1}\left(\mathrm{PL}_n/\mathrm{O}_n\right)$ for any $n\geq 1$ and to $\Omega^{n+1}\left(\mathrm{TOP}_n/\mathrm{O}_n\right)$ 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 $\mathrm{Emb}_\partial^{fr}(D^m,D^n)\simeq\Omega^{m+1}\left(\mathrm{O}_n\backslash\!\!\backslash\mathrm{PL}_n/\mathrm{PL}_{n,m}\right)\simeq \Omega^{m+1}\left(\mathrm{O}_n\backslash\!\!\backslash\mathrm{TOP}_n/\mathrm{TOP}_{n,m}\right)$ 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 $\mathrm{Emb}_\partial^{fr}(D^m,D^n)$.