Tokyo Journal of Mathematics

Some Remarks on the Coarse Index Theorem for Complete Riemannian Manifolds


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In this paper, we state two results on the coarse index for simple models of the coarse geometry. The first one is a coarse index theorem on trivial bundles over even dimensional closed Riemannian manifolds. It comes from the Poincar\'e duals for these base manifold in the trivial bundles. We next show some correspondence between the generator of the coarse cohomology of $\mathbf{R}^n$ and the Riesz transformation on $\mathbf{R}^n$.

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Tokyo J. Math., Volume 26, Number 2 (2003), 329-345.

First available in Project Euclid: 5 June 2009

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KAMIMURA, Shingo. Some Remarks on the Coarse Index Theorem for Complete Riemannian Manifolds. Tokyo J. Math. 26 (2003), no. 2, 329--345. doi:10.3836/tjm/1244208594.

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  • B. Blackadar, $K$-theory for operator algebras. Second edition, Math. Sci. Res. Inst. Publ. 5, Cambridge Univ. Press (1998).
  • R. Bott and L. W. Tu, Differential forms in algebraic topology, GTM 82, Springer (1982).
  • E. H. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Anal., 12 (1973), 401–414.
  • A. Connes, Noncommutative differential geometry, IHES, 62 (1985), 257–360.
  • A. Connes and H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, 29 (1990), 345–388.
  • H. P. McKean and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., 1 (1967), 43–69.
  • H. Moriyoshi and T. Natsume, Operator algebras and geometry, Math. Memoires of MSJ 2 (2001), 185 (in Japanese).
  • J. Roe, An index theorem on open manifolds I, J. Diff. Geom., 27 (1988), 87–113.
  • J. Roe, Partioning noncompact manifolds and the dual Toeplitz problem, Operator algebras and applications Vol. 1, London Math. Soc. Lecture Note Ser. 135 (1989) Cambridge Univ. Press, 187–228.
  • J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Memoires of AMS 497 (1993).
  • J. Roe, Index theory, coarse geometry and topology of manifolds, CBMS Regional Conference Series in Mathematics 90 (1996), AMS Providence.
  • E. H. Spanier, Algebraic topology, McGraw-Hill (1966).
  • E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press (1970).
  • M. Taylor, Pseudodifferential operators, Princeton Mathematical Series 34, (1981) Princeton Univ. Press.