Year: 2026
Type: article
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 \ \
Source: Mathematische Zeitschrift
Authors: Paolo Salvatore, Victor Turchin
Institutions: University of Rome Tor Vergata, Kansas State University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Awards: G2404429419, G5743846505, G6868696325
OpenAlex ID: W4406072153