Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-02T18:43:01.920Z Has data issue: false hasContentIssue false
  • English
  • Français

K-cycles for twisted K-homology

Published online by Cambridge University Press:  23 May 2013

Paul Baum ,
Alan Carey  and
Bai-Ling Wang
Paul Baum
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, [email protected]
Alan Carey
Affiliation:
Mathematical Sciences Institute, The Australian National University, Canberra, [email protected]
Bai-Ling Wang
Affiliation:
Mathematical Sciences Institute, The Australian National University, Canberra, [email protected]
Get access

Abstract

We summarise the construction of geometric cycles and their use in describing the Kasparov K-homology of a CW-complex X. When Kasparov K-homology is twisted by a degree three element of the Čech cohomology of X then there is a corresponding construction of twisted geometric cycles for the case where X is a smooth manifold however the method that was employed does not apply in the case of CW-complexes. In this article we propose a new approach to the construction of twisted geometric cycles for CW-complexes motivated by the study of D-branes in string theory.

Keywords

D-branestwisted K-theoryPrimary 19L5019K3519K36
Type
Research Article
Information
Journal of K-Theory , Volume 12 , Issue 1: Nanjing Special Issue on K-theory, number theory and geometry , August 2013 , pp. 69 - 98
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

1.Atiyah, M.F., K-theory, Lecture notes by D. W. Anderson, W. A. Benjamin, Inc., New York-Amsterdam 1967.Google Scholar
2.Atiyah, M. F., Global theory of elliptic operators, Proc. Internat. Conf. on Functional Analysis and Related Topics, ( Tokyo, 1969) pp. 21–30, 1970.Google Scholar
3.Baum, P., K-homology and D-branes, Superstrings, Geometry, Topology, and C*-Algebras, Proc. Symp. Pure Math. 81 (Doran, R. S., Friedman, G., and Rosenberg, J. eds.) pp. 81–94, AMS, Providence, 2010.Google Scholar
4.Baum, P., Douglas, R. G., K-homology and index theory, Operator Algebras and Applications, Part 1, (Kadison, R. ed.) Proc. Symp., Pure Math. 38, pp. 117–173, AMS, Providence, 1982.Google Scholar
5.Baum, P., Higson, N., Schick, T., On the equivalence of geometric and analytic K-homology, Pure and Applied Mathematics Quarterly 3 (2007) 1–24.CrossRefGoogle Scholar
6.Baum, P., Higson, N., Schick, T., A geometric description of equivariant K-homology for proper actions, Quanta of Maths 11, Proceedings of meeting in honor of Alain Connes' 60-th birthday, Clay Mathematics Institute, Cambridge MA, 2010.Google Scholar
7.Bott, R., Tu, L. W., Differential Forms in Algebraic Topology, Springer, 1982.CrossRefGoogle Scholar
8.Carey, A., Wang, B. L., Thom isomorphism and Push-forward map in twisted K-theory. Journal of K-Theory 1 (2008), 357–393.CrossRefGoogle Scholar
9.Conner, P., Floyd, E., Diff erentiable Periodic Maps, Springer, Lecture Notes in Mathematics 738.Google Scholar
10.Deeley, R., Geometric K-homology with coefficients I, J. K-Theory 9, 537–564 (2012); II, preprint, arXiv:1101.0697-1101.0703.Google Scholar
11.Deeley, R. J., Goffeng, M., Applying geometric K-cycles to fractional indices. preprint, arXiv:1211.1553v2 [math.KT].Google Scholar
12.Echterhoff, S., Emerson, H., and Kim, H. J., KK-theoretic duality for proper twisted actions. Math. Ann. 340 (2008) 839–873.CrossRefGoogle Scholar
13.Jakob, M., An alternative approach to homology, Contemp. Math. 265 (2000) 87–97.CrossRefGoogle Scholar
14.Kasparov, G. G., Topological invariants of elliptic operators I. K-homology, Izv. Acad. Nauk SSSR Ser. Mat. 39 (1975) 796–838 (Russian); English transl. Math. USSR- Izv. 9 (1976) 751–792.Google Scholar
15.Kasparov, G. G., The operator K-functor and extensions of C*-algebras, Izv. Acad. Nauk SSSR Ser. Mat. 44 (3) (1980), 571–636 (Russian); English transl. Math. USSR- Izv. 16 (1981) 513–572.Google Scholar
16.Kasparov, G. G., Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988) 147–201.Google Scholar
17.Kreck, M., Lück, W., The Novikov Conjecture – Geometry and Algebra, Oberwolfach Seminars, Birkhäuser, 2005.Google Scholar
18.Kuiper, N., The homotopy t ype of the unitary group of Hilbert space, Topology 3 (1965) 19–30.Google Scholar
19.Lawson, H. B., Michelsohn, M-L, Spin Geometry. Princeton University Press, 1989Google Scholar
20.Raven, J., An equivariant bivariant Chern character, PhD Thesis, Pennsylvania State University, 2004.Google Scholar
21.Schick, T., L 2-index theorems, KK-theory, and connections, New York J. Math. 11 (2005) 387–443.Google Scholar
22.Spanier, E. H., Algebraic Topology, McGraw-Hill, New York, 1980.Google Scholar
23.Tu, J-L., Twisted K-theory and Poincaré duality, Trans. Amer. Math. Soc. 361 (2009) 1269–1278.Google Scholar
24.Wang, B. L., Geometric cycles, Index theory and twisted K-homology J. Noncommut. Geom. 2 (2008) 497–552.Google Scholar
25.Whitehead, G. W., Elements of Homotopy Theory, Springer, 1978.Google Scholar