The \(h\)-principle for maps transverse to bracket-generating distributions (Q6607987)

The author partially answers a long standing question posed by \textit{M. Gromov} [Partial differential relations. Berlin etc.: Springer-Verlag (1986; Zbl 0651.53001)], see also \textit{D. McDuff}, ``Partial differential relations'', Bull. Amer. Math. Soc. 18, No. 2, 214--220 (1988)], as well as Part 2 of the excellent recent book: [\textit{K. K. Cieliebak} et al., Introduction to the \(h\)-principle. 2nd edition. Providence, RI: American Mathematical Society (AMS) (2024; Zbl 1531.58008)]. The main point concerns the proof of the fact that given a bracket-generating distribution \(D\) on a manifold \(M\) then maps from an arbitrary manifold to \(M\), which are transverse to \(D\), satisfy the \(h\)-principle. Gromov himself proposed a proof, see his book quoted above, but it turned out that his proof was valid only for specific distributions, as first pointed out by Eliashberg and Mishachev. In the paper under review the author proves Gromov's claim for smooth bracket-generating distributions of constant growth. Through the introduction of definitions 2.3 and 3.1, Theorem 3.4 (Gromov) and by providing a judicious algorithmic argument the author proves the crucial Theorem 3.7, which then allows him to implement some of Gromov's analytic and sheaf-theoretic techniques, leading to the \(h\)-principles for transverse maps (Theorem 4.1) and for transverse immersions (Theorem 4.3).