Boundary values of diffeomorphisms of simple polytopes, and controllability
Helge Glöckner, Erlend Grong, Alexander Schmeding
TL;DR
The paper develops an infinite-dimensional Lie group framework for the diffeomorphism groups of simple polytopes and analyzes their quotients by boundary-fixed diffeomorphisms. It constructs a regular Fréchet–Lie group structure on $\mathrm{Diff}^{\mathrm{fr}}(M)/\mathrm{Diff}^{\partial,\mathrm{id}}(M)$ via a smooth submersion from $\mathrm{Diff}^{\mathrm{fr}}(M)$, and proves regularity transfer results that imply $\mathrm{im}(\rho)$ is $L^1$-regular and hence $C^k$-regular for all $k$. A key technical tool is an extension operator: compatible $C^m$-data on faces extend to all of $M$ with a continuous linear right inverse, enabling a Lie-group description of the face-wise restriction image. The controllability results show that the identity component $\mathrm{Diff}(M)_0$ is generated by exponentials of stratified vector fields, and, for the subgroup fixing the boundary, by exponentials of vector fields vanishing on the boundary. Together, these results provide a robust, local-to-global understanding of diffeomorphism groups on simple polytopes with implications for boundary behavior and controllability in geometric settings.
Abstract
We consider the Lie group of smooth diffeomorphisms Diff$(M)$ of a simple polytope $M$ in the euclidean space. Simple polytopes are special cases of manifolds with corners. The geometric setting allows to study in particular, the subgroup of face respecting diffeomorphisms and its Lie theoretic properties. We find a canonical Lie group structure for the quotient of the diffeomorphism by the subgroup Diff$^{\partial,id}(M)$ of maps that equal the identity on the boundary, turning the canonical quotient homomorphism Diff$(M)\rightarrow $Diff$(M)/$Diff$^{\partial,id}(M)$ into a smooth submersion. We also show that the identity component of the diffeomorphism group is generated by the exponential image, by proving general controllability results.