Existence of Engel structures (Q2389135)

An Engel structure on a smooth 4-manifold \(M\) is a maximally non-integrable distribution \(\mathcal D\) of rank two, meaning that \(\mathcal E =[\mathcal D, \mathcal D]\) is a distribution of rank three and \([\mathcal E, \mathcal E] = TM\), where \([\mathcal D, \mathcal D]\) is defined by taking all the commutators of local sections of \(\mathcal D\). Engel structures are all locally equivalent, as are contact structures on odd-dimensional manifolds and symplectic structures on even-dimensional manifolds. The proof of the main theorem below establishes a deep connection between Engel 4-manifolds and contact 3-manifolds. A result of V. Gershkovich [see \textit{M. Kazarian}, \textit{R. Montgomery} and \textit{B. Shapiro}, Pac. J. Math. 179, No.~2, 355--370 (1997; Zbl 0895.58004)] states that every orientable Engel 4-manifold is parallelizable. In the paper under review, the author shows the converse of this, namely, every parallelizable 4-manifold is Engel. This is the main theorem. The proof is based on the existence of round handles decompositions on parallelizable 4-manifolds. A round handle of dimension \(n\) and index \(k\) is \(D^k \times D^{n-k-1} \times S^1\) attached to an \(n\)-manifold along \(S^{k-1} \times D^{n-k-1} \times S^1\) by an embedding into the boundary (for \(k=0\) there is no attachment at all). By a theorem of \textit{D. Asimov} [Ann. Math. (2) 102, 41--54 (1975; Zbl 0316.57020)], a closed manifold \(M\) of dimension \(\neq 3\) admits a round handle decomposition if and only if \(\chi(M) = 0\). By means of 3-dimensional contact geometry, the author constructs basic Engel structures on each round handle of dimension four. Also, he shows how to extend an Engel structure over a round handle attached to an Engel 4-manifold. The extended Engel structure on the round handle is equivalent to a basic one. This is the scheme of the proof. Consider a round handle decomposition of a parallelizable 4-manifold \(M\). Then, start from a suitable basic Engel structure on the round handle of index zero. By the extension procedure, the author is able to get an Engel structure on \(M\).