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

Alexandr Mishchenko

Open Mathematics (2005)

  • Volume: 3, Issue: 4, page 766-793
  • ISSN: 2391-5455

Abstract

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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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Alexandr Mishchenko. "K-theory from the point of view of C*-algebras and Fredholm representations." Open Mathematics 3.4 (2005): 766-793. <http://eudml.org/doc/268820>.

@article{AlexandrMishchenko2005,
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.},
author = {Alexandr Mishchenko},
journal = {Open Mathematics},
keywords = {19L; 19K; 19J25; 55N15},
language = {eng},
number = {4},
pages = {766-793},
title = {K-theory from the point of view of C*-algebras and Fredholm representations},
url = {http://eudml.org/doc/268820},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Alexandr Mishchenko
TI - K-theory from the point of view of C*-algebras and Fredholm representations
JO - Open Mathematics
PY - 2005
VL - 3
IS - 4
SP - 766
EP - 793
AB - 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.
LA - eng
KW - 19L; 19K; 19J25; 55N15
UR - http://eudml.org/doc/268820
ER -

References

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  1. [1] M.A. Naimark: Normed algebras, Nauka, Moscow, 1968 (in Russian), English translation: Wolters-NoordHoff, 1972. 
  2. [2] A. Connes: Noncommutative Geometry. Academic Press, 1994. 
  3. [3] L.S. Pontryagin: “Classification of some fiber bundles”, Dokl. Akad. Nauk SSSR, Vol. 47(5), (1945), pp. 327–330 (in Russian). 
  4. [4] L.S. Pontryagin: “Characteristic cycles of differntiable manifolds”, Mathem. Sbornik, Vol. 21(2), (1947), pp 233–284 (in Russian). 
  5. [5] W. Browder: Homotopy type of differential manifolds, Colloquium on Algebraic Topology, Aarhus Universitet, Aarhus, Matematisk Institut, 1962, pp. 42–46. 
  6. [6] S.P. Novikov: “Homotopy equivalent smoth manifolds, I”, Izv. Akad. Nauk SSSR, Ser. Mat., Vol. 28, (1964), pp. 365–474. Zbl0151.32103
  7. [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. http://dx.doi.org/10.1073/pnas.39.9.951 Zbl0051.14501
  8. [8] V.A. Rokhlin: “New results in the theory of 4-dimensional manifolds”, Dokl. Akad. Nauk SSSR, Vol. 84, (1952), pp. 221–224. 
  9. [9] A. Poincare: “Analysis situs”, Journal de l’Ecole Polytechniques, Vol. 1, (1895), pp. 1–121. Zbl26.0541.07
  10. [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. 
  11. [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. Zbl0957.58021
  12. [12] A.S. Mishchenko: “ C *-algebras and K-theory”, Algebraic topology, Lecutre Notes in Mathematics, Vol. 763, Aarhus, 1979, pp 262–274. http://dx.doi.org/10.1007/BFb0088090 
  13. [13] C.T.C. Wall, Surgery of compact manifolds, Academic Press, New York, 1971. 
  14. [14] R. Thom: “Quelques properietes globales des varietes differentiables” Comm. Math. Helv., Vol. 28, (1954), pp. 17–86. Zbl0057.15502
  15. [15] G. Luke and A.S. Mishchenko: Vector Bundles and Their Applications, Kluwer Academic Publishers, Dordrrecht, Boston, London, 1998. 
  16. [16] D. Husemoller: Fiber Bundles, McGraw-Hill, N.Y., 1966. 
  17. [17] M. Karoubi: K-Theory, An Introduction, Springer-Verlag, Berlin, Heidelberg, new York, 1978. 
  18. [18] M.F. Atiah: “ k-theory and reality”, Quart. J. Math. Oxford, Ser., (2), Vol. 17, (1966), pp.367–386. Zbl0146.19101
  19. [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. 
  20. [20] R. Palais: Seminar on the atiyah-singer index theorem, Ann. of Math. Stud., Vol. 57, Princeton Univ. Press, Princeton, N.J., 1965. 
  21. [21] W. Paschke: “Inner product modules over b *-algebras” Trans. Amer. Math. Soc., Vol. 182, (1973), pp. 443–468. http://dx.doi.org/10.2307/1996542 Zbl0239.46062
  22. [22] N. Kuiper: “The homotopy type of the unitary group of hilbert space”, Topology, Vol. 3, (1965), pp.19–30. http://dx.doi.org/10.1016/0040-9383(65)90067-4 
  23. [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. Zbl0416.46052
  24. [24] G. Kasparov: “Equivariant kk-theory and the novikov conjecture”, Invent. Math., Vol. 91, (1988), pp. 147–201. http://dx.doi.org/10.1007/BF01404917 Zbl0647.46053
  25. [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. Zbl1163.47032
  26. [26] M.F. Atiyah and G. Segal: “Twisted k-theory”, arXiv: math.KT 0407054 vl, 5Jul2004. 
  27. [27] A.S. Mishchenko: “Theory of almost algebraic Poincaré complexes and local combinatorial Hirzebruch formula”, Acta. Appl. Math., Vol. 68, (2001), pp. 5–37. http://dx.doi.org/10.1023/A:1012062214272 
  28. [28] G. Lusztig: “Novikov’s higher signature and families of elliptic operators”” J. Diff. Geometry, Vol. 7, (1972), pp. 229–256. Zbl0265.57009
  29. [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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