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K-theory from the point of view of C *-algebras and Fredholm representations

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Central European Journal of Mathematics

Abstract

These notes represent the subject of five lectures which were delivered as a minicourse during the VI conference in Krynica, Poland, “Geometry and Topology of Manifolds”, May, 2–8, 2004.

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References

  1. M.A. Naimark: Normed algebras, Nauka, Moscow, 1968 (in Russian), English translation: Wolters-NoordHoff, 1972.

    Google Scholar 

  2. A. Connes: Noncommutative Geometry. Academic Press, 1994.

  3. L.S. Pontryagin: “Classification of some fiber bundles”, Dokl. Akad. Nauk SSSR, Vol. 47(5), (1945), pp. 327–330 (in Russian).

    MathSciNet  Google Scholar 

  4. L.S. Pontryagin: “Characteristic cycles of differntiable manifolds”, Mathem. Sbornik, Vol. 21(2), (1947), pp 233–284 (in Russian).

    MATH  MathSciNet  Google Scholar 

  5. W. Browder: Homotopy type of differential manifolds, Colloquium on Algebraic Topology, Aarhus Universitet, Aarhus, Matematisk Institut, 1962, pp. 42–46.

  6. S.P. Novikov: “Homotopy equivalent smoth manifolds, I”, Izv. Akad. Nauk SSSR, Ser. Mat., Vol. 28, (1964), pp. 365–474.

    MathSciNet  Google Scholar 

  7. F. Hirzebruch: “On Steenrod’s reduced powers the index of interia, and the Todd genus”, Proc. Nat., Acad. Sci. U.S.A., Vol. 39, (1953), pp. 951–956.

    Article  MATH  MathSciNet  Google Scholar 

  8. V.A. Rokhlin: “New results in the theory of 4-dimensional manifolds”, Dokl. Akad. Nauk SSSR, Vol. 84, (1952), pp. 221–224.

    MATH  Google Scholar 

  9. A. Poincare: “Analysis situs”, Journal de l’Ecole Polytechniques, Vol. 1, (1895), pp. 1–121.

    MATH  MathSciNet  Google Scholar 

  10. A.S. Mishchenko: “Homotopy invariant of non simply connected manifolds. Rational invariants. I”, Izv. Akad. Nauk. SSSR, Vol. 43(3), (1970), pp. 501–514.

    Google Scholar 

  11. A.S. Mishchenko: “Controlled Fredholm representations”, In: S.C. Ferry, A. Ranicki and J. Rosenberg (Eds.): Novikov Conjectures, Index Theorems and Rigidity Vol. 1, London Math. Soc. Lect. Notes Series, Vol, 226, Cambridge Univ. Press, 1995, pp. 174–200.

  12. A.S. Mishchenko: “ C *-algebras and K-theory”, Algebraic topology, Lecutre Notes in Mathematics, Vol. 763, Aarhus, 1979, pp 262–274.

    Article  MATH  MathSciNet  Google Scholar 

  13. C.T.C. Wall, Surgery of compact manifolds, Academic Press, New York, 1971.

    Google Scholar 

  14. R. Thom: “Quelques properietes globales des varietes differentiables” Comm. Math. Helv., Vol. 28, (1954), pp. 17–86.

    MATH  MathSciNet  Google Scholar 

  15. G. Luke and A.S. Mishchenko: Vector Bundles and Their Applications, Kluwer Academic Publishers, Dordrrecht, Boston, London, 1998.

    MATH  Google Scholar 

  16. D. Husemoller: Fiber Bundles, McGraw-Hill, N.Y., 1966.

    Google Scholar 

  17. M. Karoubi: K-Theory, An Introduction, Springer-Verlag, Berlin, Heidelberg, new York, 1978.

    MATH  Google Scholar 

  18. M.F. Atiah: “ k-theory and reality”, Quart. J. Math. Oxford, Ser., (2), Vol. 17, (1966), pp.367–386.

    MathSciNet  Google Scholar 

  19. M.F. Atiyah: “Global theory of elliptic operators”, In: Intern. Conf. on Functional Analysis and Related Topics (Tokyo 1969), Univ. Tokyo Press, Tokyo, 1970, pp. 21–30.

    Google Scholar 

  20. R. Palais: Seminar on the atiyah-singer index theorem, Ann. of Math. Stud., Vol. 57, Princeton Univ. Press, Princeton, N.J., 1965.

    MATH  Google Scholar 

  21. W. Paschke: “Inner product modules over b *-algebras” Trans. Amer. Math. Soc., Vol. 182, (1973), pp. 443–468.

    Article  MATH  MathSciNet  Google Scholar 

  22. N. Kuiper: “The homotopy type of the unitary group of hilbert space”, Topology, Vol. 3, (1965), pp.19–30.

    Article  MATH  MathSciNet  Google Scholar 

  23. A.S. Mishchenko and A.T. Fomenko: “Index of elliptic operators over C *-algebras”, Izvrstia AN SSSR, ser. matem., Vol. 43(4), (1979), pp. 831–859.

    MATH  MathSciNet  Google Scholar 

  24. G. Kasparov: “Equivariant kk-theory and the novikov conjecture”, Invent. Math., Vol. 91, (1988), pp. 147–201.

    Article  MATH  MathSciNet  Google Scholar 

  25. A.A. Irmatov and A.S. Mishchenko: “On compact and fredholm operators over c*-algebras and a new topology in the space of compact operators”, arXiv:math.KT 0504548 vl 27 Apr 2005.

  26. M.F. Atiyah and G. Segal: “Twisted k-theory”, arXiv: math.KT 0407054 vl, 5Jul2004.

  27. A.S. Mishchenko: “Theory of almost algebraic Poincaré complexes and local combinatorial Hirzebruch formula”, Acta. Appl. Math., Vol. 68, (2001), pp. 5–37.

    Article  MATH  MathSciNet  Google Scholar 

  28. G. Lusztig: “Novikov’s higher signature and families of elliptic operators”” J. Diff. Geometry, Vol. 7, (1972), pp. 229–256.

    MATH  MathSciNet  Google Scholar 

  29. S.C. Ferry, A. Ranicki and J. Rosenberg (Eds.): Proceedings of the Conference ’Novikov Conjectures, Index Theorems and Rigidity, Cambrdige Univ. Press, 1995.

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Authors and Affiliations

  1. Departments of Mathematics and Mechanics, Moscow State University, Leninskije Gory, 119899, Moscow, Russia

    Alexandr S. Mishchenko

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  1. Alexandr S. Mishchenko
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Mishchenko, A.S. K-theory from the point of view of C *-algebras and Fredholm representations. centr.eur.j.math. 3, 766–793 (2005). https://doi.org/10.2478/BF02475629

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  • DOI: https://doi.org/10.2478/BF02475629

Keywords

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