Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T12:42:51.100Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric K-homology with coefficients I: ℤ/kℤ-cycles and Bockstein sequence

Published online by Cambridge University Press:  04 November 2011

Robin J. Deeley
Robin J. Deeley
Affiliation:
Mathematisches Institut, Georg-August Universität, Bunsenstrasse 3-5, 37073 Göttingen, Germanyrjdeeley@uni-math.gwdg.de
Get access

Abstract

We construct a Baum-Douglas type model for K-homology with coefficients in ℤ/kℤ. The basic geometric object in a cycle is a spinc ℤ/kℤ-manifold. The relationship between these cycles and the topological side of the Freed-Melrose index theorem is discussed in detail. Finally, using inductive limits, we construct geometric models for K-homology with coefficients in any countable abelian group.

Keywords

K-homologygeometric cyclesℤ/kℤ-manifoldsindex theory
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 3 , June 2012 , pp. 537 - 564
Copyright
Copyright © ISOPP 2011

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.. Global theory of elliptic operators. Proc. of the International Sym. on Functional Analysis, Tokyo, 1970. University of Tokyo Press.Google Scholar
2.Atiyah, M. F., Patodi, V. K., and Singer, I. M.. Spectral asymmetry and Riemannian geometry I, Math. Proc. Camb. Phil. Soc. 77: 43–69, 1975.CrossRefGoogle Scholar
3.Atiyah, M. F., Patodi, V. K., and Singer, I. M.. Spectral asymmetry and Riemannian geometry II, Math. Proc. Camb. Phil. Soc. 78: 405–432, 1975.CrossRefGoogle Scholar
4.Atiyah, M. F., Patodi, V. K., and Singer, I. M.. Spectral asymmetry and Riemannian geometry III, Math. Proc. Camb. Phil. Soc. 79: 71–99, 1976.CrossRefGoogle Scholar
5.Atiyah, M. and Singer, I., The index of elliptic operators, I. Ann. of Math. 87: 531–545, 1968.CrossRefGoogle Scholar
6.Baas, N. A.. On bordism theory of manifolds with singularities. Math. Scand. 33: 279–302, 1974.CrossRefGoogle Scholar
7.Baum, P. and Douglas, R.. K-homology and index thoery. Operator Algebras and Applications (Kadison, R. editor), Proceedings of Symposia in Pure Math. 38: 117–173, Providence RI, 1982. AMS.Google Scholar
8.Baum, P. and Douglas, R.. Index theory, bordism, and K-homology. Contemp. Math. 10:1–31, 1982.CrossRefGoogle Scholar
9.Baum, P. and Douglas, R.. Relative K-homology and C*-algebras, K-Theory 5: 1–46, 1991.CrossRefGoogle Scholar
10.Baum, P., Douglas, R., and Taylor, M.. Cycles and relative cycles in analytic K-homology. J. Diff. Geo. 30: 761–804, 1989.Google Scholar
11.Baum, P., Higson, N., and Schick, T.. On the equivalence of geometric and analytic K-homology. Pure Appl. Math. Q. 3: 1–24, 2007CrossRefGoogle Scholar
12.Brown, L., Douglas, R., and Fillmore, P.. Unitary equivalence modulo the compact operators and extensions of C*-algebras. Proc. Conf. Operator Theory (Dalhousie Univ., Halifax, N.S., 1973), Lecture Notes in Mathematics, Springer 345: 58–128, 1973.CrossRefGoogle Scholar
13.Brown, L., Douglas, R., and Fillmore, P.. Extensions of C*-algebras and K-homology. Annals of Math. 105: 265–324, 1977.CrossRefGoogle Scholar
14.Conner, P. and Floyd, E.. The relation of cobordism to K-theories. Lecture Notes in Math. 725, Springer-Verlag, 1966.Google Scholar
15.Connes, A. and Skandalis, G.. The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. 20, no. 6: 1139–1183 1984.CrossRefGoogle Scholar
16.Deeley, R.. Geometric K-homology with coefficients II. arXiv: 1101.07.03v1.Google Scholar
17.Emerson, H. and Meyer, R.. Bivariant K-theory via correspondences. Adv. Math. 225, no. 5: 2883–2919, 2010.CrossRefGoogle Scholar
18.Freed, D. S.. ℤ/k-manifolds and families of Dirac operators. Invent. Math. 92: 243–254, 1988.CrossRefGoogle Scholar
19.Freed, D. S. and Melrose, R. B.. A mod k index theorem. Invent. Math. 107: 283–299, 1992.CrossRefGoogle Scholar
20.Higson, N.. An approach to ℤ/k-index theory. Internat. J. Math. 1: 283–299, 1990.CrossRefGoogle Scholar
21.Higson, N. and Roe, J.. Analytic K-homology. Oxford University Press, Oxford, 2000.Google Scholar
22.Higson, N. and Roe, J.. K-homology, assembly and rigidity theorems for relative eta invariants. Pure Appl. Math. Q. 6, no. 2, Special Issue: In honor of Michael Atiyah and Isadore Singer, 555–601 (2010).CrossRefGoogle Scholar
23.Hirsch, M.. Differential topology. Springer-Verlag, 1976.CrossRefGoogle Scholar
24.Jakob, M.. A bordism-type construction of homology. Manuser. Math. 96: 67–80 1998.CrossRefGoogle Scholar
25.Kasparov, G. G.. Topological invariants of elliptic operators I: K-homology. Math. USSR Izvestija 9:751–792, 1975.CrossRefGoogle Scholar
26.Kaminker, J. and Wojciechowski, K. P.. Index theory of ℤ/k manifolds and the Grassmannian, Operator Algebras and Topology (Craiova, 1989), Longman, Harlow, 82–92, 1992.Google Scholar
27.Morgan, J. W. and Sullivan, D. P.. The transversality charactersitic class and linking cycles in surgery theory, Annals of Math. 99: 463–544, 1974.CrossRefGoogle Scholar
28.Raven, J.. An equivariant bivariant chern character, PhD Thesis, Pennsylvania State University, 2004. (available online at the Pennsylvania State Digital Library).Google Scholar
29.Rordam, M., Larsen, F., and Laustsen, N.. An Introduction to K-theory for C*-algebras. London Math. Soc. Student Text 49, Cambridge University Press, 2000Google Scholar
30.Rosenberg, J.. Groupoid C*-algebras and index theory on manifolds with singularities, Geom. Dedicata 100: 5–84, 2003.CrossRefGoogle Scholar
31.Schochet, C.. Topological methods for C*-algebras IV: mod p homology, Pacific Journal of Math. 114: 447–468.CrossRefGoogle Scholar
32.Sullivan, D. P.. Geometric topology, part I: Localization, periodicity and Galois symmetry, MIT, 1970Google Scholar
33.Sullivan, D. P.. Triangulating and smoothing homotopy equivalences and homeomorphims: geometric topology seminar notes, The Hauptvermuting Book, Kluwer Acad. Publ., Dordrecht, 69–103, 1996.CrossRefGoogle Scholar
34.Zhang, W.. On the mod k index theorem of Freed and Melrose, J. Diff. Geo. 43: 198–206, 1996.Google Scholar