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\(AF\)-algebras and topology of mapping tori. (English) Zbl 1374.46054

Summary: The paper studies applications of \(C^*\)-algebras in geometric topology. Namely, a covariant functor from the category of mapping tori to a category of \(AF\)-algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding \(AF\)-algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension 2, 3 and 4. In conclusion, we consider two numerical examples illustrating our main results.

MSC:

46L85 Noncommutative topology
55S35 Obstruction theory in algebraic topology
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[1] D. V. Anosov: Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov. 90 (1967), 209. (In Russian.)
[2] M. Bauer: A characterization of uniquely ergodic interval exchange maps in terms of the Jacobi-Perron algorithm. Bol. Soc. Bras. Mat., Nova Sér. 27 (1996), 109–128. · Zbl 0877.11044 · doi:10.1007/BF01259355
[3] L. Bernstein: The Jacobi-Perron Algorithm. Its Theory and Application. Lecture Notes in Mathematics 207, Springer, Berlin, 1971. · Zbl 0213.05201
[4] O. Bratteli: Inductive limits of finite dimensional C*-algebras. Trans. Am. Math. Soc. 171 (1972), 195–234. · Zbl 0264.46057
[5] E. G. Effros: Dimensions and C*-Algebras. Regional Conference Series in Mathematics 46, Conference Board of the Mathematical Sciences, Washington, AMS, Providence, 1981.
[6] H. B. Lawson, Jr.: Foliations. Bull. Am. Math. Soc. 80 (1974), 369–418. · Zbl 0293.57014 · doi:10.1090/S0002-9904-1974-13432-4
[7] P. Morandi: Field and Galois Theory. Graduate Texts in Mathematics 167, Springer, New York, 1996. · Zbl 0865.12001
[8] J. F. Plante: Foliations with measure preserving holonomy. Ann. Math. (2) 102 (1975), 327–361. · Zbl 0314.57018 · doi:10.2307/1971034
[9] S. Smale: Differentiable dynamical systems. Bull. Am. Math. Soc. 73 (1967), 747–817. · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1
[10] W. P. Thurston: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc., New Ser. 19 (1988), 417–431. · Zbl 0674.57008 · doi:10.1090/S0273-0979-1988-15685-6
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