Euler structures, nonsingular vector fields, and torsions of Reidemeister type.

*(English. Russian original)*Zbl 0692.57015
Math. USSR, Izv. 34, No. 3, 627-662 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 3, 607-643 (1989).

There is a natural definition of the set Eul(X) of all Euler structures on a polyhedron X. The set of all homology classes of non-singular vector fields on a smooth manifold M is denoted by vect(M). Two vector fields \(u_ 0\) and \(u_ 1\) are homologous if they cn be included in some family \(u_ t\) with finitely many singularities. The main result is the construction of a canonical isomorphism Eul(M)\(\to vect(M)\) for smooth manifolds M of dimension \(\geq 2\). As a consequence the following is obtained: if two diffeomorphisms f,g: \(M\to N\) are homotopic then \(Df=Dg: vect(M)\to vect(N)\). It is proved that in the case of odd-dimensional manifolds some torsions of Reidemeister type \(\theta\) (M,u) are full invariants of Euler structures. At the end of the paper there is a nice application of these results to classification of spin structures on 3- manifolds.

Reviewer: A.Dranishnikov

##### MSC:

57R25 | Vector fields, frame fields in differential topology |

57R20 | Characteristic classes and numbers in differential topology |

57R15 | Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |