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

• Autorzy: Alexandr Mishchenko
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Vector bundle; K-theory; C*-algebras; Fredholm operators; Fredholm representations; 19L; 19K; 19J25; 55N15

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2005
• Tom: 3
• Numer: 4
• Strony: 766-793

Daty

wydano

2005-12-01

online

2005-12-01

Twórcy

autor

Alexandr Mishchenko

• Moscow State University

Bibliografia

• [1] M.A. Naimark: Normed algebras, Nauka, Moscow, 1968 (in Russian), English translation: Wolters-NoordHoff, 1972.
• [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).
• [4] L.S. Pontryagin: “Characteristic cycles of differntiable manifolds”, Mathem. Sbornik, Vol. 21(2), (1947), pp 233–284 (in Russian).
• [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.
• [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
• [8] V.A. Rokhlin: “New results in the theory of 4-dimensional manifolds”, Dokl. Akad. Nauk SSSR, Vol. 84, (1952), pp. 221–224.
• [9] A. Poincare: “Analysis situs”, Journal de l’Ecole Polytechniques, Vol. 1, (1895), pp. 1–121.
• [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] 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. http://dx.doi.org/10.1007/BFb0088090
• [13] C.T.C. Wall, Surgery of compact manifolds, Academic Press, New York, 1971.
• [14] R. Thom: “Quelques properietes globales des varietes differentiables” Comm. Math. Helv., Vol. 28, (1954), pp. 17–86.
• [15] G. Luke and A.S. Mishchenko: Vector Bundles and Their Applications, Kluwer Academic Publishers, Dordrrecht, Boston, London, 1998.
• [16] D. Husemoller: Fiber Bundles, McGraw-Hill, N.Y., 1966.
• [17] M. Karoubi: K-Theory, An Introduction, Springer-Verlag, Berlin, Heidelberg, new York, 1978.
• [18] M.F. Atiah: “ k-theory and reality”, Quart. J. Math. Oxford, Ser., (2), Vol. 17, (1966), pp.367–386.
• [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] R. Palais: Seminar on the atiyah-singer index theorem, Ann. of Math. Stud., Vol. 57, Princeton Univ. Press, Princeton, N.J., 1965.
• [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
• [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] 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.
• [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
• [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. http://dx.doi.org/10.1023/A:1012062214272
• [28] G. Lusztig: “Novikov’s higher signature and families of elliptic operators”” J. Diff. Geometry, Vol. 7, (1972), pp. 229–256.
• [29] S.C. Ferry, A. Ranicki and J. Rosenberg (Eds.): Proceedings of the Conference ’Novikov Conjectures, Index Theorems and Rigidity, Cambrdige Univ. Press, 1995.

Identyfikatory

DOI

10.2478/BF02475629