Relative H-Principle and Contact Geometry

TL;DR

The paper develops a relative h-principle framework for spaces of holonomic versus formal solutions and proves that, under a sufficiently separated collection of relative data, the holonomic-to-formal map admits a section up to homotopy. This machinery is then specialized to contact geometry, showing that the overtwisted h-principle yields a homotopy section for Cont$^{OT}(M) \to$ AlmCont$(M)$ and implying that $\pi_k$ AlmCont$(M)$ embeds in $\pi_k$ Cont$^{OT}(M)$ for all $k$ (with the image matching $OT_k(M)$ in the stable range $k\le 2n$). These results are leveraged to produce infinite cyclic subgroups in the rational homotopy of the contactomorphism group of closed overtwisted manifolds, via a rational-injectivity analysis of disk diffeomorphisms and their induced action. The framework further extends to Engel geometry, yielding analogous section results and inviting comparisons with existing subgroups arising from loose Engel structures. Overall, the work provides a unified, homotopy-theoretic approach to comparing holonomic and formal structures across contact and Engel settings, with precise implications for the topology of diffeomorphism and contactomorphism groups.

Abstract

We show that if $F(M)$ is some space of holonomic solutions with space of formal solutions $F^f(M)$ that satisfies a certain relative $h$-principle, then the non-relative map $F(M) \to F^f(M)$ admits a section up to homotopy. We apply this to the relative $h$-principle for overtwisted contact structures proved by Borman-Eliashberg-Murphy to find infinite cyclic subgroups in the homotopy groups of the contactomorphism group of $M$.

Theorems & Definitions (34)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Lemma 2.5
• proof
• Theorem 2.6
• proof
• Remark 2.7