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Algebraic v. Topological K-Theory: A Friendly Match

Part of the Lecture Notes in Mathematics book series (LNM,volume 2008)

Abstract

These notes evolved from the lecture notes of a minicourse given in Swisk, the Sedano Winter School on K theory held in Sedano, Spain, during the week January 22 to 27 of 2007, and from those of a longer course given in the University of Buenos Aires, during the second half of 2006.

Keywords

  • Exact Sequence
  • Banach Algebra
  • Ring Homomorphism
  • Unital Ring
  • Cyclic Homology

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Anderson, D.: Relationship among K-theories. In: Bass, H. (ed.) Higher K-theories. Lecture Notes in Mathematics, vol. 341, pp. 57–72. Springer, New York (1972)

    Google Scholar 

  2. Bass, H.: Algebraic K-theory. W.A. Benjamin, New York (1968)

    Google Scholar 

  3. Benson, D.J.: Representations and cohomology. In: Cambridge Studies in Advanced Mathematics, Second edition, vol. 30. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  4. Berrick, A.J.: An approach to algebraic K-theory. In: Research Notes in Mathematics, vol. 56. Pitman Books, London (1982)

    Google Scholar 

  5. Blackadar, B.: K-theory for operator algebras, Second edition. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  6. Bousfield, A., Kan, D.: Homotopy limits, completions and localizations. In: Lecture Notes in Mathematics, vol. 304. Springer, Berlin (1972)

    Google Scholar 

  7. Bousfield, A., Friedlander, E.: Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. In: Geometric Applications of Homotopy Theory (Proc. Conf., Evanston, Ill., 1977), II. Lecture Notes in Mathematics, vol. 658, pp. 80–130. Springer, Berlin (1978)

    Google Scholar 

  8. Browder, W.: Higher torsion in H-spaces. Trans. Am. Math. Soc. 108, 353–375 (1963)

    MATH  MathSciNet  Google Scholar 

  9. Connes, A., Karoubi, M.: Caractére mulitplicatif d’un module de Fredholm. K-theory 2, 431–463 (1988)

    Google Scholar 

  10. Cortiñas, G.: The obstruction to excision in K-theory and in cyclic homology. Invent. Math. 454, 143–173 (2006)

    CrossRef  Google Scholar 

  11. Cortiñas, G, Thom, A.: Bivariant algebraic K-theory. J. reine angew. Math. 510, 71–124 (2007)

    Google Scholar 

  12. Cortiñas, G., Thom, A.: Comparison between algebraic and topological K-theory of locally convex algebras. math.KT/067222

    Google Scholar 

  13. Cortiñas, G., Valqui, C.: Excision in bivariant periodic cyclic homology: a categorical approach. K-theory 30, 167–201 (2003)

    Google Scholar 

  14. Cuntz, J.: K-theory and C -algebras. In: Algebraic K-theory, number theory and analysis. Lecture Notes in Mathematics, vol. 1046, pp. 55–79. Springer, Berlin (1984)

    Google Scholar 

  15. Cuntz, J.: Bivariante K-theorie für lokalkonnvexe Algebren und der Chern-Connes-Charakter. Doc. Math. 2, 139–182 (1997)

    MATH  MathSciNet  Google Scholar 

  16. Cuntz, J.: Bivariant K-theory and the Weyl algebra. K-Theory 35, 93–137 (2005)

    Google Scholar 

  17. Cuntz, J., Quillen, D.: Cyclic homology and nonsingularity. J. Am. Math. Soc. 8, 373–441 (1995)

    MATH  MathSciNet  Google Scholar 

  18. Cuntz, J., Quillen, D.: Excision in periodic bivariant cyclic cohomology. Invent. Math. 127, 67–98 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. Cuntz, J., Thom, A.: Algebraic K-theory and locally convex algebras. Math. Ann. 334, 339–371 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Davidson, K. R.: C -algebras by example. In: Fields Institute Monographs, vol. 6. American Mathematical Society, Providence (1996)

    Google Scholar 

  21. Gabriel, P., Zisman, G.: Calculus of fractions and homotopy theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer, New York (1967)

    Google Scholar 

  22. Gelfand, I., Manin, Y.: Methods of homological algebra. In: Springer Monographs in Mathematics, second edition. Springer, New York (2002)

    Google Scholar 

  23. Grayson, D.: Higher algebraic K-theory II. (after Daniel Quillen). Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976). In: Lecture Notes in Mathematics, vol. 551, pp. 217–240. Springer, Berlin (1976)

    Google Scholar 

  24. Goerss, P., Jardine, J.: Simplicial homotopy theory. In: Progress in Mathematics, vol. 174. Birkhuser, Basel (1999)

    Google Scholar 

  25. Goodwillie, T.: Cyclic homology, derivations, and the free loopspace. Topology 24, 187–215 (1985)

    CrossRef  MATH  MathSciNet  Google Scholar 

  26. Goodwillie, T.: Relative algebraic K-theory and cyclic homology. Ann. Math. 124, 347–402 (1986)

    CrossRef  MathSciNet  Google Scholar 

  27. Higson, N.: Algebraic K-theory ofC -algebras. Adv. Math. 67, 1–40 (1988)

    CrossRef  Google Scholar 

  28. Hirschorn, P.: Model categories and their localizations. In: Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Providence (2003)

    Google Scholar 

  29. Husemöller, D.: Algebraic K-theory of operator ideals (after Mariusz Wodzicki). In: K-theory, Strasbourg 1992. Astérisque 226, 193–209 (1994)

    Google Scholar 

  30. Karoubi, M.: K-théorie algébrique de certaines algèbres d’opérateurs. In: Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978). Lecture Notes in Mathematics, vol. 725, pp. 254–290. Springer, Berlin (1979)

    Google Scholar 

  31. Karoubi, M.: Homologie cyclique et K-théorie. Astérisque, Société Mathématique de France 149 (1987)

    Google Scholar 

  32. Karoubi, M.: Sur la K-théorie Multiplicative. Cyclic homology and noncommutative geometry. Fields Inst. Commun. 17, 59–77 (1997)

    MathSciNet  Google Scholar 

  33. Karoubi, M., Villamayor, O.: K-théorie algebrique et K-théorie topologique. Math. Scand. 28, 265–307 (1971)

    MATH  MathSciNet  Google Scholar 

  34. Loday, J.L.: Cyclic homology, 1st edition. Grundlehren der mathematischen Wis- senschaften, vol. 301. Springer, Berlin (1998)

    Google Scholar 

  35. Loday, J.L.: K-théorie algébrique et représentations de groupes. Ann. Sci. ENS. 4 Série, tome 9(3), 309–377 (1976)

    Google Scholar 

  36. Milnor, J.: Introduction to algebraic K-theory. In: Annals of Mathematics Studies, vol. 72. Princeton University Press, Princeton (1971)

    Google Scholar 

  37. Phillips, N.C.: K-theory for Fréchet algebras. Int. J. Math. 2, 77–129 (1991)

    CrossRef  MATH  Google Scholar 

  38. Pirashvili, T.I.: Some properties of Karoubi-Villamayor algebraic K-functors (Russian) Soobshch. Akad. Nauk Gruzin. SSR 97(2), 289–292 (1980)

    MATH  MathSciNet  Google Scholar 

  39. Quillen, D.: Higher algebraic K-theory. I. Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972). In: Lecture Notes in Mathematics, vol. 341, pp. 85–147. Springer, Berlin (1973)

    Google Scholar 

  40. Quillen, D.: Higher algebraic K-theory. In: Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), vol. 1, pp. 171–176. Canad. Math. Congress, Montreal, Que. (1975)

    Google Scholar 

  41. Rordam, M., Larsen, F., Laustsen, N.J.: An introduction to K-theory for C -algebras. In: London Mathematical Society Student Texts, vol. 49. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  42. Rosenberg, J.: Algebraic K-theory and its applications. In: Graduate Texts in Mathematics, vol. 147. Springer, New York (1994)

    Google Scholar 

  43. Rosenberg, J.: Comparison between algebraic and topological K-theory for Banach algebras and C -algebras. In: Friedlander, E.M., Grayson, D.R. (eds.) Handbook of K-Theory. Springer, New York (2005)

    Google Scholar 

  44. Schlichting, M.: Higher Algebraic K-Theory II (After Quillen, Thomason and Others). This volume

    Google Scholar 

  45. Summers, M.: Factorization in Frchet modules. J. London Math. Soc. 5, 243–248 (1972)

    CrossRef  MATH  MathSciNet  Google Scholar 

  46. Suslin, A.: Excision in the integral algebraic K-theory. Proc. Steklov Inst. Math. 208, 255–279 (1995)

    MathSciNet  Google Scholar 

  47. Suslin, A., Wodzicki, M.: Excision in algebraic K-theory. Ann. Math. 136, 51–122 (1992)

    CrossRef  MathSciNet  Google Scholar 

  48. Swan, R.: Excision in algebraic K-theory. J. Pure Appl. Algebra 1, 221–252 (1971)

    CrossRef  MATH  MathSciNet  Google Scholar 

  49. Switzer, R.: Algebraic topology-homotopy and homology. Grundlehren der mathematischen Wis- senschaften, vol.212. Springer, Berlin (1975)

    Google Scholar 

  50. Vorst, T.: Localization of the K-theory of polynomial extensions. Math. Ann. 244, 33–54 (1979)

    CrossRef  MATH  MathSciNet  Google Scholar 

  51. Wagoner, J.B.: Delooping classifying spaces in algebraic K-theory. Topology 11, 349–370 (1972)

    CrossRef  MATH  MathSciNet  Google Scholar 

  52. Wegge-Olsen, N.E.:K-theory andC -algebras. Oxford University Press, Oxford (1993)

    Google Scholar 

  53. Weibel, C.: The K-book: An introduction to algebraic K-theory. Book-in-progress, available at its author’s webpage: http://www.math.rutgers.edu/~weibel/Kbook.html

  54. Weibel, C.: An introduction to homological algebra. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  55. Weibel, C.: Homotopy Algebraic K-theory. Contemp. Math. 83, 461–488 (1989)

    MathSciNet  Google Scholar 

  56. Weibel, C.: Nil K-theory maps to cyclic homology. Trans. Am. Math. Soc. 303, 541–558 (1987)

    MATH  MathSciNet  Google Scholar 

  57. Weibel, C.A.: Nilpotence and K-theory. J. Algebra 61(2), 298–307 (1979)

    CrossRef  MATH  MathSciNet  Google Scholar 

  58. Whitehead, G.W.: Elements of Homotopy Theory. Springer, Berlin (1978)

    MATH  Google Scholar 

  59. Wodzicki, M.: Excision in cyclic homology and in rational algebraic K-theory. Ann. Math. (2) 129, 591–639 (1989)

    Google Scholar 

  60. Wodzicki, M.: Algebraic K-theory and functional analysis. In: First European Congress of Mathematics, Vol. II (Paris, 1992). Progress in Mathematics, vol. 120, pp. 485–496. Birkhäuser, Basel (1994)

    Google Scholar 

  61. Zeeman, E.C.: A proof of the comparison theorem for spectral sequences. Proc. Camb. Philos. Soc. 53, 57–62 (1957)

    CrossRef  MATH  MathSciNet  Google Scholar 

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Acknowledgements.

Work for these notes was partly supported by FSE and by grants PICT03-12330, UBACyT-X294, VA091A05, and MTM00958.

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Correspondence to Guillermo Cortiñas .

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Cortiñas, G. (2011). Algebraic v. Topological K-Theory: A Friendly Match. In: Topics in Algebraic and Topological K-Theory. Lecture Notes in Mathematics(), vol 2008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15708-0_3

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