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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)
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