Diffeomorphism groups of compact locally polyhedral manifolds

Abstract: A locally polyhedral manifold $M$ is a manifold which is locally diffeomorph to open subsets of convex polytopes. Manifolds with corners are a special case of locally polyhedral manifolds. We turn the group $\text{Diff}^{\infty}(M)$ of all smooth diffeomorphisms $f\colon M\to M$ for compact smooth $M$ into a regular Lie group.

• The paper establishes a smooth, regular Lie group structure on Diff∞(M) using stratified vector fields.
• It embeds compact locally polyhedral manifolds into boundaryless manifolds to extend standard C∞ analysis with local additions and sprays.
• The construction resolves gaps in earlier approaches to manifolds with corners, offering new insights for geometric and infinite-dimensional analysis.

Diffeomorphism Groups of Compact Locally Polyhedral Manifolds

Overview and Main Results

This paper rigorously develops the infinite-dimensional Lie group structure of the diffeomorphism group $\mathrm{Diff}^{\infty}(M)$, where $M$ is a compact locally polyhedral $C^{\infty}$-manifold. Locally polyhedral manifolds generalize both manifolds with corners and open subsets of polytopes, providing a flexible yet tractable geometric setting. The central contribution is the construction and verification of a smooth, regular Lie group structure on $\mathrm{Diff}^{\infty}(M)$, modelled on the space of stratified smooth vector fields. The authors bridge gaps left open in previous discussions on manifolds with boundary and corners, and provide a new, comprehensive framework for this broader class.

Background and Definitions

A locally polyhedral manifold is a Hausdorff space equipped with an atlas whose charts map open subsets of $M$ onto open subsets of convex polytopes in a finite-dimensional vector space, with $C^{\infty}$ chart transitions. Manifolds with corners (e.g., domains in $[0,1]^n$) are special cases. The diffeomorphism group $\mathrm{Diff}^{k}(M)$ consists of $C^k$ invertible self-maps with $C^k$ inverses.

A central technical aspect is modelling $M$0 not on the full space of vector fields, but on the closed subspace $M$1 of stratified vector fields, which respect the stratification induced by the faces of local polytopes (i.e., vector fields whose values on any $M$2-codimensional stratum are tangent to that stratum).

Lie Group Structure Construction

Embedding into a Boundaryless Manifold

A foundational step is embedding $M$3 as a locally polyhedral full submanifold of a $M$4-compact, finite-dimensional smooth manifold $M$5 without boundary. This is accomplished via flows of strictly inner vector fields on $M$6, ensuring $M$7 is realized as a submanifold where the extension of $M$8 analysis is standard and sprays with suitable properties can be constructed.

Stratified Vector Fields and Manifold Modelling

The manifold structure on $M$9 is modelled on the stratified vector field space, which retains Banach or Fréchet structure (depending on $C^{\infty}$0 or $C^{\infty}$1) due to the closedness of this subspace. This modelling is shown to be necessary to ensure local charts behave well near the stratified boundary/corner behavior of $C^{\infty}$2.

Local Additions and Exponential Law

A smooth local addition (exponential chart) associated with a suitably adapted spray on $C^{\infty}$3 is employed to define smooth charts on $C^{\infty}$4. Open neighborhoods in $C^{\infty}$5 are mapped homeomorphically onto open sets in $C^{\infty}$6, with a further restriction yielding charts for $C^{\infty}$7 itself.

The exponential law is established for this structure: a map $C^{\infty}$8 from any $C^{\infty}$9-manifold $\mathrm{Diff}^{\infty}(M)$0 (possibly with rough boundary) is $\mathrm{Diff}^{\infty}(M)$1 if and only if the induced map $\mathrm{Diff}^{\infty}(M)$2 is $\mathrm{Diff}^{\infty}(M)$3, ensuring compatibility with higher categorical and functional analytic constructions.

Regularity and Group Operations

The group structure (composition, inversion) is verified to be smooth, and the group is shown to be $\mathrm{Diff}^{\infty}(M)$4-regular in the sense of Glöckner and, hence, regular in the sense of Milnor. This is done via explicit calculations on the effect of composition and inversion in charts adapted to stratified vector fields, as well as by referencing strong ILB–Lie group theory and suitable projective limit arguments.

Special Case: Convex Polytopes

When $\mathrm{Diff}^{\infty}(M)$5 is a convex polytope, the group of face-respecting diffeomorphisms is characterized and shown to be an open subset of the full diffeomorphism group. The local addition and stratified vector field formalism specialize to explicit linear-algebraic conditions on tangent vectors and vector fields at the faces of $\mathrm{Diff}^{\infty}(M)$6. The analysis establishes the equivalence between stratifiedness and invariance of faces under diffeomorphisms—a core structural property leveraged throughout the general theory.

Implications

The explicit Lie group structure on $\mathrm{Diff}^{\infty}(M)$7 for compact locally polyhedral manifolds resolves longstanding technical issues—most notably, those exposed in previous work by Michor for manifolds with corners. By showing that stratified vector fields provide the correct modeling space, the paper establishes a rigorous, general template for the infinite-dimensional Lie theory of diffeomorphism groups in settings relevant for geometric analysis, infinite-dimensional geometry, and subfields such as analysis on stratified spaces.

The result directly enables the development of geometric analysis (e.g., flows, group actions, moduli) on stratified types of manifolds ubiquitous in global analysis, manifold theory for singular spaces, and mathematical physics. Practically, the regularity properties assure applicability of Lie group techniques—such as flows and evolution equations—even in the presence of corners and non-smooth boundaries.

Future Directions

Potential future directions include:

• Extending these constructions to noncompact or infinite-dimensional polyhedral contexts, possibly with additional structure (e.g., fiber bundles, groupoids).
• Analyzing the fine homotopy and homology properties of $\mathrm{Diff}^{\infty}(M)$8 for various $\mathrm{Diff}^{\infty}(M)$9, particularly as they relate to the stratification.
• Investigating the interplay between this regular Lie group structure and symplectic or Poisson structures on locally polyhedral manifolds.
• Applying this framework to problems in gauge theory or geometric quantization on manifolds with boundary/corners.

Conclusion

This work provides a comprehensive and rigorous Lie group structure for diffeomorphism groups of compact locally polyhedral smooth manifolds, resolving technical obstacles and establishing a general blueprint for infinite-dimensional Lie theory in singular geometric settings. The central role of stratified vector fields and the careful construction of local additions and sprays are highlighted as both necessary and sufficient for the desired regular infinite-dimensional Lie group properties, with broad implications for global analysis and geometry on singular and stratified spaces.