Hostname: page-component-7dd5485656-7jgsp Total loading time: 0 Render date: 2025-10-30T19:41:23.021Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric K-homology with coefficients II: The Analytic Theory and Isomorphism

Published online by Cambridge University Press:  28 August 2013

Robin J. Deeley
Get access

Abstract

We discuss the analytic aspects of the geometric model for K-homology with coefficients in ℤ/kℤ constructed in [12]. In particular, using results of Rosenberg and Schochet, we construct a map from this geometric model to its analytic counterpart. Moreover, we show that this map is an isomorphism in the case of a finite CW-complex. The relationship between this map and the Freed-Melrose index theorem is also discussed. Many of these results are analogous to those of Baum and Douglas in the case of spinc manifolds, geometric K-homology, and Atiyah-Singer index theorem.

Keywords

K-homologyℤ/kℤ-manifoldsthe Freed-Melrose index theoremPrimary: 19K33Secondary: 19K5646L8555N20

Information

Type
Research Article
Information
Journal of K-Theory , Volume 12 , Issue 2 , October 2013 , pp. 235 - 256
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.)

Article purchase

Temporarily unavailable

References

1.Atiyah, M. F., Patodi, V. K., and Singer, I. M.. Spectral asymmetry and Riemannian geometry I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43–69.CrossRefGoogle Scholar
2.Atiyah, M. and Singer, I., The index of elliptic operators, I. Ann. of Math. 87 (1968), 531–545.Google Scholar
3.Baum, P. and Douglas, R.. K-homology and index theory. Operator Algebras and Applications (R. Kadison editor), Proceedings of Symposia in Pure Math. 38, 117–173, Providence RI, 1982. AMS.Google Scholar
4.Baum, P. and Douglas, R.. Index theory, bordism, and K-homology. Contemp. Math. 10 (1982), 1–31.CrossRefGoogle Scholar
5.Baum, P. and Douglas, R.. Relative K-homology and C*-algebras, K-Theory 5 (1991), 1–46.Google Scholar
6.Baum, P., Douglas, R., and Taylor, M.. Cycles and relative cycles in analytic K-homology. J. Diff. Geo. 30 (1989), 761–804.Google Scholar
7.Baum, P., Higson, N., and Schick, T.. On the equivalence of geometric and analytic K-homology. Pure Appl. Math. Q. 3 (2007), 1–24.CrossRefGoogle Scholar
8.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 345, Springer, 1973, pp 58–128.Google Scholar
9.Brown, L., Douglas, R., and Fillmore, P.. Extensions of C*-algebras and K-homology. Annals of Math. 105 (1977), 265–324.CrossRefGoogle Scholar
10.Conner, P. and Floyd, E.. Differentiable periodic maps. Springer-Verlag, 1964.Google Scholar
11.Dai, X. and Zhang, W.. An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary. Journal of Func. Anal. 238 (2006), 1–26.Google Scholar
12.Deeley, R. J.. Geometric K-homology with coefficients I: ℤ/kℤ-cycles and Bockstein sequence. J. K-Theory 9(3) (2012), 537–564.Google Scholar
13.Freed, D. S. and Melrose, R. B.. A mod k index theorem. Invent. Math. 107 (1992), 283–299.Google Scholar
14.Higson, N.. An approach to ℤ/k-index theory. Internat. J. Math. 1 (1990), 283–299.Google Scholar
15.Higson, N. and Roe, J.. Analytic K-homology. Oxford University Press, Oxford, 2000.Google Scholar
16.Higson, N., Roe, J., and Rosenberg, J.. Introduction to non-commutative geometry, Notes based on lectures from the Clay Inst. Sym. at Mt. Holyoke College 2000.Google Scholar
17.Kasparov, G. G.. Topological invariants of elliptic operators I: K-homology. Math. USSR Izvestija 9 (1975), 751–792.Google Scholar
18.Morgan, J. W. and Sullivan, D. P.. The transversality characteristic class and linking cycles in surgery theory, Annals of Math. 99 (1974), 463–544.Google Scholar
19.Renault, J.. A groupoid approach to C*-algebras, Springer, Berlin, 1980.Google Scholar
20.Rosenberg, J.. Groupoid C *-algebras and index theory on manifolds with singularities, Geom. Dedicata 100 (2003), 65–84.Google Scholar
21.Schochet, C.. Topological methods for C*-algebras IV: mod p homology, Pacific Journal of Math. 114, 447–468.Google Scholar
22.Sullivan, D. P.. Geometric topology, part I: Localization, periodicity and Galois symmetry, MIT, 1970.Google Scholar
23.Sullivan, D. P.. Triangulating and smoothing homotopy equivalences and homeomor-phims: geometric topology seminar notes, The Hauptvermuting Book, Kluwer Acad. Publ., Dordrecht, 69–103, 1996.Google Scholar