Remarks on flat and differential K-theory
[Remarques sur les K-théories plate et différentielle]
Ho, Man-Ho
Annales mathématiques Blaise Pascal, Tome 21 (2014), p. 91-101 / Harvested from Numdam
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Dans cette note, nous prouvous certains résultats en K-théories plate et différentielle. La premier est une preuve de la compatibilité de l’indice topologique différentiel et de l’indice topologique plat par un calcul direct. Le second est un isomorphisme explicite entre les K-théories différentielles de Bunke-Schick et de Freed-Lott.

In this note we prove some results in flat and differential K-theory. The first one is a proof of the compatibility of the differential topological index and the flat topological index by a direct computation. The second one is the explicit isomorphisms between Bunke-Schick differential K-theory and Freed-Lott differential K-theory.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/ambp.337
Classification:  19L50,  58J20
Mots clés: différentielle K-théorie, indice topologique 
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     title = {Remarks on flat and differential $K$-theory},
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     year = {2014},
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     doi = {10.5802/ambp.337},
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Ho, Man-Ho. Remarks on flat and differential $K$-theory. Annales mathématiques Blaise Pascal, Tome 21 (2014) pp. 91-101. doi : 10.5802/ambp.337. http://gdmltest.u-ga.fr/item/AMBP_2014__21_1_91_0/

[1] Baum, P.; Higson, N.; Schick, T. On the equivalence of geometric and analytic K-homology, Pure Appl. Math. Q., Tome 3 (2007) no. 1, part 3, pp. 1-24 | Article | MR 2330153 | Zbl 1146.19004

[2] Bismut, J.M.; Zhang, W. Real embeddings and eta invariants, Math. Ann., Tome 295 (1993) no. 4, pp. 661-684 | Article | MR 1214954 | Zbl 0795.57010

[3] Bunke, U. Index theory, eta forms, and Deligne cohomology, Mem. Amer. Math. Soc., Tome 198 (2009) no. 928, pp. vi+120 | MR 2191484 | Zbl 1181.58017

[4] Bunke, U.; Schick, T. Smooth K-theory, Astérisque (2009) no. 328, pp. 45-135 | MR 2664467 | Zbl 1202.19007

[5] Bunke, U.; Schick, T. Uniqueness of smooth extensions of generalized cohomology theories, J. Topol., Tome 3 (2010) no. 1, pp. 110-156 | Article | MR 2608479 | Zbl 1252.55002

[6] Bunke, U.; Schick, T.; Bär, C.; Lohkamp, J.; Schwarz, M. Differential K-theory. A survey, Global Differential Geometry, Springer-Verlag, Berlin Heidelberg (Springer Proceedings in Mathematics) Tome 17 (2012), pp. 303-358 | Zbl 1245.19002

[7] Freed, D.; Lott, J. An index theorem in differential K-theory, Geom. Topol., Tome 14 (2010) no. 2, pp. 903-966 | Article | MR 2602854 | Zbl 1197.58007

[8] Ho, M.-H. The differential analytic index in Simons-Sullivan differential K-theory, Ann. Global Anal. Geom., Tome 42 (2012) no. 4, pp. 523-535 | Article | MR 2995203 | Zbl 1257.19005

[9] Hopkins, M. J.; Singer, I. M. Quadratic functions in geometry, topology, and M-theory, J. Differential Geom., Tome 70 (2005) no. 3, pp. 329-452 | MR 2192936 | Zbl 1116.58018

[10] Klonoff, K. An index theorem in differential K -theory, The University of Texas at Austin (2008), pp. 119 (Ph. D. Thesis) | MR 2711943

[11] Lott, J. ℝ/ℤ index theory, Comm. Anal. Geom., Tome 2 (1994) no. 2, pp. 279-311 | MR 1312690 | Zbl 0840.58044

[12] Simons, J.; Sullivan, D. Structured vector bundles define differential K-theory, Quanta of maths, Amer. Math. Soc., Providence, RI (Clay Math. Proc.) Tome 11 (2010), pp. 579-599 | MR 2732065 | Zbl 1216.19009