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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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