# Article

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Keywords:
Anosov diffeomorphism; \$AF\$-algebra
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.
References:
[1] Anosov, D. V.: Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov. 90 Russian (1967), 209. MR 0224110
[2] Bauer, M.: 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. DOI 10.1007/BF01259355 | MR 1418928 | Zbl 0877.11044
[3] Bernstein, L.: The Jacobi-Perron Algorithm. Its Theory and Application. Lecture Notes in Mathematics 207 Springer, Berlin (1971). DOI 10.1007/BFb0069405 | MR 0285478 | Zbl 0213.05201
[4] Bratteli, O.: Inductive limits of finite dimensional {\$C^{\ast} \$}-algebras. Trans. Am. Math. Soc. 171 (1972), 195-234. MR 0312282
[5] Effros, E. G.: Dimensions and \$C^\ast \$-Algebras. Regional Conference Series in Mathematics 46 Conference Board of the Mathematical Sciences, Washington, AMS, Providence (1981). MR 0623762
[6] H. B. Lawson, Jr.: Foliations. Bull. Am. Math. Soc. 80 (1974), 369-418. DOI 10.1090/S0002-9904-1974-13432-4 | MR 0343289 | Zbl 0293.57014
[7] Morandi, P.: Field and Galois Theory. Graduate Texts in Mathematics 167 Springer, New York (1996). DOI 10.1007/978-1-4612-4040-2 | MR 1410264 | Zbl 0865.12001
[8] Plante, J. F.: Foliations with measure preserving holonomy. Ann. Math. (2) 102 (1975), 327-361. MR 0391125 | Zbl 0314.57018
[9] Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73 (1967), 747-817. DOI 10.1090/S0002-9904-1967-11798-1 | MR 0228014 | Zbl 0202.55202
[10] Thurston, W. P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc., New Ser. 19 (1988), 417-431. DOI 10.1090/S0273-0979-1988-15685-6 | MR 0956596 | Zbl 0674.57008

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