Euler structures, nonsingular vector fields, and torsions of Reidemeister type. (English. Russian original) Zbl 0692.57015

Turaev, V. G.

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)

Keywords:

Reidemeister torsion; homotopic diffeomorphism; set of all Euler structures on a polyhedron; set of all homology classes of non-singular vector fields; spin structures on 3-manifolds