Boundary values of diffeomorphisms of simple polytopes, and controllability

TL;DR

This paper studies the structure of the diffeomorphism group of simple polytopes, establishing a Lie group framework for the quotient by boundary-fixing diffeomorphisms and demonstrating controllability of the identity component.

Contribution

It introduces a canonical Lie group structure on the quotient of the diffeomorphism group and proves controllability results for the identity component.

Findings

Canonical Lie group structure on the quotient of Diff(M)

The quotient map is a smooth submersion

The identity component is generated by the exponential map

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.