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A coarse Mayer–Vietoris principle

Published online by Cambridge University Press:  24 October 2008

Nigel Higson ,
John Roe  and
Guoliang Yu
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
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 85 - 97
Copyright
Copyright © Cambridge Philosophical Society 1993

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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), 337–341.CrossRefGoogle Scholar
[3]Pedersen, E. K. and Weibel, C. A.. A nonconnective delooping of algebraic K-theory. Springer Lecture Notes in Mathematics 1126 (1985), 306–320.Google Scholar
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[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