##
**Existence of Engel structures.**
*(English)*
Zbl 1195.58005

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 M. Kazarian, R. Montgomery and 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 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 \(M\).

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 M. Kazarian, R. Montgomery and 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 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 \(M\).

Reviewer: Daniele Zuddas (Cagliari)

### MSC:

58A30 | Vector distributions (subbundles of the tangent bundles) |

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

53D10 | Contact manifolds (general theory) |