Remarks on flat and differential K-theory
Annales mathématiques Blaise Pascal, Volume 21 (2014) no. 1, pp. 91-101.

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.

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.

DOI: 10.5802/ambp.337
Classification: 19L50, 58J20
Keywords: differential $K$-theory, topological index
Mot clés : différentielle $K$-théorie, indice topologique

Man-Ho Ho 1

1 Department of Mathematics Hong Kong Baptist University Kowloon Tong, Kowloon Hong Kong
@article{AMBP_2014__21_1_91_0,
     author = {Man-Ho Ho},
     title = {Remarks on flat and differential $K$-theory},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {91--101},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {21},
     number = {1},
     year = {2014},
     doi = {10.5802/ambp.337},
     mrnumber = {3248223},
     zbl = {06329058},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.337/}
}
TY  - JOUR
AU  - Man-Ho Ho
TI  - Remarks on flat and differential $K$-theory
JO  - Annales mathématiques Blaise Pascal
PY  - 2014
SP  - 91
EP  - 101
VL  - 21
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.337/
DO  - 10.5802/ambp.337
LA  - en
ID  - AMBP_2014__21_1_91_0
ER  - 
%0 Journal Article
%A Man-Ho Ho
%T Remarks on flat and differential $K$-theory
%J Annales mathématiques Blaise Pascal
%D 2014
%P 91-101
%V 21
%N 1
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.337/
%R 10.5802/ambp.337
%G en
%F AMBP_2014__21_1_91_0
Man-Ho Ho. Remarks on flat and differential $K$-theory. Annales mathématiques Blaise Pascal, Volume 21 (2014) no. 1, pp. 91-101. doi : 10.5802/ambp.337. https://ambp.centre-mersenne.org/articles/10.5802/ambp.337/

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

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

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

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

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

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

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

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

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

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

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

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

Cited by Sources: