Wrinkling and Haefliger structures

TL;DR

The paper develops a comprehensive wrinkling framework for Diff-invariant differential relations, recasting Haefliger structures as a natural setting for holonomic approximation into étale spaces of solutions ${\operatorname{EtSol}}_{\mathcal{R}}(M)$. It extends wrinkling from submersions to arbitrary order, proving parametric and relative versions that lift formal data to wrinkled holonomic sections, and showing that the projection to the solution space is highly connected. Central to the approach is the notion of ${\mathcal{R}}$-microbundles, a Haefliger-like generalization that encodes transverse ${\mathcal{R}}$-geometry on microbundles and can be deformed to genuine ${\mathcal{R}}$-structures via wrinkling; this is extended to tangential cases and to general dimensions. The paper then builds a classifying framework using groupoids ${\Gamma}_{\mathcal{R}}$ and their formal analogues, establishing connectivity results for the scanning map and for the inclusion of genuine into formal bundles, with applications to folded symplectic forms and horizontal homotopy groups. Collectively, the results provide a robust, Diff-invariant h-principle architecture that unifies wrinkling, étale spaces, and Haefliger-type groupoids, enabling broad transfers of formal data to genuine geometric structures with controlled singularities.

Abstract

Wrinkling techniques, introduced by Eliashberg and Mishachev, are typically used to prove h-principles of the form: ``formal solutions of a partial differential relation $\mathcal{R}$ can be deformed to singular/wrinkled solutions''. What a wrinkled solution is depends on the context, but the overall idea is that it should be an object that fails to be a solution only due to the presence of mild/controlled singularities. Much earlier, Haefliger structures were introduced by Haefliger as singular analogues of foliations. Much like a foliation is locally modeled on a submersion, a Haefliger structure is modeled on an arbitrary map. This implies that Haefliger structures have better formal properties than foliations. For instance, they can be pulled back by arbitrary maps and admit a classifying space. In [12], the second and third authors generalized the \emph{wrinkled embeddings} of Eliashberg and Mishachev to arbitrary order. This paper can be regarded as a sequel in which we deal instead with generalizations of \emph{wrinkled submersions}. The main messages are that: 1) Haefliger structures provide a nice conceptual framework in which general wrinkling statements can be made. 2) Wrinkling can be interpreted as holonomic approximation into the étale space of solutions of the relation $\mathcal{R}$. These statements imply connectivity statements relating (1) $\mathcal{R}$ to its étale space of solutions and (2) the classifying space for foliations with transverse $\mathcal{R}$-geometry to its formal counterpart.

Figures (6)

• Figure 1: Illustrations of the cusp.
• Figure 2: Locus of singularities of a wrinkle. We have indicated what singularity we obtain when restricted to the indicated domain.
• Figure 3: A sketch of wrinkling map $G_\ell$: on the left we see its domain $I^n$ with the domains of four $s_y$. The grey ovals indicate where the wrinkling takes place. On the right we see the image of $G_\ell$ in ${J^\textrm{germs}} \Psi$. We have drawn the wrinkles there in the color of the strip in which they take place. The dotted lines indicate the interpolation in between the strips. Note that the wrinkles do not overlap in the domain, but they do in the target when projected to $M$.
• Figure 4: On the left, the image of a map $F: S^i \to {\mathcal{R}}$. The fibers of the bundle ${\mathcal{R}} \to M$ run vertically. In the middle, a piecewise embedding of $S^i$ into ${\mathcal{R}}$ that is transverse to the fibers and a $C^0$-approximation of $F$. On the right, a "holonomic approximation" of $F$ by a map projected down from étale space. Along each top simplex we see wrinkling.
• Figure 5: A depiction of the proof of \ref{['thm:wrinklingMicro']}. On the left, we see the étale space ${{{\operatorname{EtSol}}}}_{{\mathcal{R}}}(M)$ lying over $M$. The zig-zag shown is the map $G$ produced by \ref{['thm:wrinklingEtale']}. When restricted to the image of $G$, the tautological ${\operatorname{\Gamma}}_{{\mathcal{R}}}$-cocycle over ${{{\operatorname{EtSol}}}}_{{\mathcal{R}}}(M)$ defines a tautological ${\mathcal{R}}$-microbundle, which we imagine as a pullback of $TM$ foliated by the fibers of the exponential map. These fibers are drawn as the little diagonal segments in turquoise. We pullback this ${\mathcal{R}}$-microbundle to $M$ itself, via the map $G$. Since $G$ is wrinkled, the pullback is not regular but instead exhibits wrinkled singularities, as depicted on the right.

Theorems & Definitions (173)

• Theorem 1.0
• Corollary 1.1
• Remark 1.2
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Proposition 1.4
• Definition 1.5
• Remark 1.6
• Definition 1.7