Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-27T03:56:51.862Z Has data issue: false hasContentIssue false

A coarse Mayer–Vietoris principle

Published online by Cambridge University Press:  24 October 2008

Nigel Higson
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802
John Roe
Affiliation:
Jesus College, Oxford, OX1 3DW
Guoliang Yu
Affiliation:
Department of Mathematics, University of Colorado, Boulder, CO 80309

Extract

In [1], [4], and [6] the authors have studied index problems associated with the ‘coarse geometry’ of a metric space, which typically might be a complete noncompact Riemannian manifold or a group equipped with a word metric. The second author has introduced a cohomology theory, coarse cohomology, which is functorial on the category of metric spaces and coarse maps, and which can be computed in many examples. Associated to such a metric space there is also a C*-algebra generated by locally compact operators with finite propagation. In this note we will show that for suitable decompositions of a metric space there are Mayer–Vietoris sequences both in coarse cohomology and in the K-theory of the C*-algebra. As an application we shall calculate the K-theory of the C*-algebra associated to a metric cone. The result is consistent with the calculation of the coarse cohomology of the cone, and with a ‘coarse’ version of the Baum–Connes conjecture.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1993

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

REFERENCES

[1]Higson, N.. On the relative K-homology theory of Baum and Douglas. J. Functional Anal., to appear.Google Scholar
[2]Milnor, J.. On axiomatic homology theory. Pacific J. Math. 12 (1962), 337341.CrossRefGoogle Scholar
[3]Pedersen, E. K. and Weibel, C. A.. A nonconnective delooping of algebraic K-theory. Springer Lecture Notes in Mathematics 1126 (1985), 306320.Google Scholar
[4]Roe, J.. Coarse cohomology theory and index theory on complete Riemannian manifolds. Memoirs Amer. Math. Soc., to appear.Google Scholar
[5]Spanier, E.. Algebraic topology (McGraw-Hill, 1966).Google Scholar
[6] Yu, G.. K-theoretic indices of Dirac type operators on complete manifolds and the Roe algebra. PhD Thesis, SUNY at Stony Brook, 1991.Google Scholar