Abstract
We provide a finite-dimensional model of the twisted K-group twisted by any degree three integral cohomology class of a CW complex. One key to the model is Furuta’s generalized vector bundle, and the other is a finite-dimensional approximation of Fredholm operators.
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Atiyah, M.F.: K-theory. Lecture notes by D. W. Anderson, New York-Amsterdam: W. A. Benjamin, Inc., 1967
Atiyah, M.F., Hirzebruch, F.: Vector bundles and homogeneous spaces. Proc. Sympos. Pure Math., Vol. III Providence, R.I.: Amer. Math. Soc., 1961, pp. 7–38
Atiyah, M.F., Segal, G.: Twisted K-theory. Ukr. Mat. Visn. 1(3), 287–330 (2004); translation in Ukr. Math. Bull. 1(3), 291–334 (2004)
Atiyah M.F., Singer I.M.: Index theory for skew-adjoint Fredholm operators. Inst. Hautes Études Sci. Publ. Math. No. 37, 5–26 (1969)
Baas, N.A., Dundas, B.I., Rognes, J.: Two-vector bundles and forms of elliptic cohomology. In: Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., 308, Cambridge: Cambridge Univ. Press, 2004, pp. 18–45
Bouwknegt P., Carey A.L., Mathai V., Murray M.K., Stevenson D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228(1), 17–45 (2002)
Brylinski, J.-L.: Categories of vector bundles and Yang-Mills equations. Contemp. Math. 230, Providence, RI: Amer. Math. Soc., 1998
Carey A.L., Hannabuss K.C., Mathai V., McCann P.: Quantum Hall effect on the hyperbolic plane. Commun. Math. Phys. 190(3), 629–673 (1998)
Carey A.L., Wang B.-L.: Thom isomorphism and Push-forward map in twisted K-theory. J. K-Theory 1(2), 357–393 (2008)
Donovan P., Karoubi M.: Graded Brauer groups and K-theory with local coefficients. Inst. Hautes Études Sci. Publ. Math. No. 38, 5–25 (1970)
Freed, D.S., Hopkins, M.J., Teleman, C.: Loop groups and twisted K-theory I. http://arXiv.org/abs/0711.1906v1[math.AT], 2007
Freed, D.S., Hopkins, M.J., Teleman, C.: Twisted K-theory and loop group representations. http://arXiv.org/abs/math/0312155v3[math.AT], 2005
Furuta M.: Index theorem, II. (Japanese) Iwanami Series in Modern Mathematics. Iwanami Shoten, Tokyo (2002)
Furuta M.: Monopole equation and the \({\frac{11}8}\)-conjecture. Math. Res. Lett. 8(3), 279–291 (2001)
Gomi, K.: An approach toward a finite-dimensional definition of twisted K-theory. Travaux mathématiques. Vol. XVII, Fac. Sci. Technol. Commun., Luxembourg: Univ. Luxemb., 2007, pp. 75–85
Gomi, K., Terashima, Y.: Chern-Weil construction for twisted K-theory. preprint
Kapustin A.: D-branes in a topologically nontrivial B-field. Adv. Theor. Math. Phys. 4(1), 127–154 (2000)
Milnor J.: On axiomatic homology theory. Pacific J. Math. 12, 337–341 (1962)
Rosenberg J.: Continuous-trace algebras from the bundle theoretic point of view. J. Austral. Math. Soc. Ser. A 47(3), 368–381 (1989)
Witten, E.: D-branes and K-theory. J. High Energy Phys. 1998(12), Paper 19, 41 pp. (1998) (electronic)
Acknowledgements
I benefited a lot from discussion with M. Furuta at various stages of the work, and I am grateful to him. I thank T. Moriyama for suggestions regarding the proof of Theorem 1. I also thank A. Henriques for suggestions about the proof and for comments on a draft. I am indebted to D. Freed and B-L. Wang for useful discussions. The author’s research is supported by a JSPS Postdoctoral Fellowship for Research Abroad.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Gomi, K. Twisted K-Theory and Finite-Dimensional Approximation. Commun. Math. Phys. 294, 863–889 (2010). https://doi.org/10.1007/s00220-009-0971-5
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DOI: https://doi.org/10.1007/s00220-009-0971-5