Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-12T19:42:02.827Z Has data issue: false hasContentIssue false
  • English
  • Français

An elementary differential extension of odd K-theory

Published online by Cambridge University Press:  04 April 2013

Thomas Tradler ,
Scott O. Wilson  and
Mahmoud Zeinalian
Get access

Abstract

There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.

Keywords

K-theorydifferential K-theoryChern-Simons19L5058J2819E20
Type
Research Article
Information
Journal of K-Theory , Volume 12 , Issue 2 , October 2013 , pp. 331 - 361
Copyright
Copyright © ISOPP 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

AP.Aguilar, M.A., Pieto, C., Quasifibrations and Bott periodicity, Topol. and its Appl. 98 (1999), 3–17.Google Scholar
B.Behrens, M., A new proof of Bott periodicity theorem, Topol. and its Appl. 119 (2002), 167–183.Google Scholar
BS.Bunke, U., Schick, T., Smooth K-Theory, Astérisque 328 (2009), 45–135.Google Scholar
BS2.Bunke, U., Schick, T., Differential K-theory: a survey, Bär, Christian, (ed.) et al., Global differential geometry. Berlin: Springer (ISBN 978-3-642-22841-4/hbk; 978-3-642-22842-1/ebook). Springer Proc. in Math. 17 (2012), 303–357.Google Scholar
BS3.Bunke, U., Schick, T., Uniqueness of smooth extensions of generalized cohomology theories, J. Topol. 3 (1) (2010), 110–156.CrossRefGoogle Scholar
C.Chen, Kuo-Tsai, Iterated path integrals, Bull. AMS 83 (1977), 831–879.Google Scholar
ChS.Cheeger, J., Simons, J.. Differential characters and geometric invariants, Geometry and topology (College Park, Md., 1983/84), 50–80, Lecture Notes in Math. 1167, Springer, Berlin, 1985.CrossRefGoogle Scholar
CS.Chern, S.S. and Simons, James. Characteristic Forms and Geometric Invariants, Ann. Math. 99(1) (1974), 48–69.Google Scholar
D.Deligne, P.. Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.Google Scholar
FL.Freed, D. D., Lott, J., An Index Theorem in Differential K-Theory, Geom. Topol. 14(2) (2010), 903–966.Google Scholar
G.Getzler, E., The odd chern character in cyclic homology and spectral flow, Topol. 32(3) (1993), 489–507.CrossRefGoogle Scholar
GJP.Getzler, E., Jones, J. D. S., Petrack, S., Differential forms on loop spaces and the cyclic bar complex, Topol. 30(3) (1991), 339–371.Google Scholar
HS.Hopkins, M.J., Singer, I.M., Quadratic functions in Geometry, Topology, and M-theory, J. Diff. Geom 70 (2005), 329–452.Google Scholar
L.Lott, J., R/Z Index Theory, Comm. Anal. Geom. 2 (1994), 279–311.Google Scholar
LT.Lundell, A., Tosa, Y., Explicit construction of nontrivial elements for homotopy groups of classical Lie groups, J. Math. Phys. 31 1494 (1990).Google Scholar
McD.McDuff, D., Configuration spaces, K-Theory and Operator Algebras, Morrel, B.B., Singer, I.M. (Eds.), Lect. Notes in Math. 575, Springer, Berlin, (1977), 88–95.Google Scholar
M.Moser, J., On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294.Google Scholar
NR.Narasimhan, M.S., Ramanan, S., Existence of universal connections. II, Amer. J. Math. 85 (1963), 223–231.CrossRefGoogle Scholar
SS.Simons, J., Sullivan, D., Structured vector bundles define differential K-Theory, Quanta of maths 579–599, Clay Math. Proc. 11, Amer. Math. Soc., Providence, RI, 2010.Google Scholar
TWZ.Tradler, T., Wilson, S.O., Zeinalian, M., Equivariant holonomy for bundles and abelian gerbes, Comm. in Math. Physics 315(1) (2012), 38–108.Google Scholar
TWZ2.Tradler, T., Wilson, S.O., Zeinalian, M., Loop differential K-theory, preprint arxiv:1201.4593.Google Scholar
Zh.Zhang, W., Lectures on Chern-Weil theory and Witten deformations. (English summary). Nankai Tracts in Math. 4, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. xii+117 pp.Google Scholar