Some remarks on the coarse index theorem for complete Riemannian manifolds. (English) Zbl 1053.58009
Coarse geometry describes Riemannian manifolds on very large scales. In this theory, coarse cohomology and coarse index theory play the role of singular cohomology and classical index theory; both deal with objects that are invariant under altering the underlying manifolds on small scales. The author computes the coarse index of untwisted Dirac operators in two special cases. Let \(M\) be a compact manifold, let \(N=M\times\mathbb{R}^n\) and let \(D_M\) and \(D_N\) be corresponding untwisted Dirac operators. Then the first result says that the coarse index of \(D_N\) equals \(c_n\operatorname {ind}(D_M)\) for some constant \(c_n>0\). The second result computes the coarse index of \(\mathbb{R}^n\) by relating the total Riesz transformation of \(L^2(\mathbb{R}^n)\) to a generator of the coarse cohomology group \(HX^n(\mathbb{R}^n)\).
Reviewer: Sebastian Goette (Regensburg)
MSC:
| 58J22 | Exotic index theories on manifolds |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
References:
| [1] | B. Blackadar, \(K\)-theory for operator algebras. Second edition , Math. Sci. Res. Inst. Publ. 5 , Cambridge Univ. Press (1998). · Zbl 0913.46054 |
| [2] | R. Bott and L. W. Tu, Differential forms in algebraic topology , GTM 82 , Springer (1982). · Zbl 0496.55001 |
| [3] | E. H. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Anal., 12 (1973), 401-414. · Zbl 0263.35066 · doi:10.1016/0022-1236(73)90003-7 |
| [4] | A. Connes, Noncommutative differential geometry, IHES, 62 (1985), 257-360. · Zbl 0592.46056 |
| [5] | A. Connes and H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, 29 (1990), 345-388. · Zbl 0759.58047 · doi:10.1016/0040-9383(90)90003-3 |
| [6] | H. P. McKean and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., 1 (1967), 43-69. · Zbl 0198.44301 |
| [7] | H. Moriyoshi and T. Natsume, Operator algebras and geometry , Math. Memoires of MSJ 2 (2001), 185 (in Japanese). · Zbl 1163.46001 |
| [8] | J. Roe, An index theorem on open manifolds I, J. Diff. Geom., 27 (1988), 87-113. · Zbl 0657.58041 |
| [9] | 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. |
| [10] | J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds , Memoires of AMS 497 (1993). · Zbl 0780.58043 |
| [11] | J. Roe, Index theory, coarse geometry and topology of manifolds , CBMS Regional Conference Series in Mathematics 90 (1996), AMS Providence. · Zbl 0853.58003 |
| [12] | E. H. Spanier, Algebraic topology , McGraw-Hill (1966). · Zbl 0145.43303 |
| [13] | E. Stein, Singular integrals and differentiability properties of functions , Princeton Univ. Press (1970). · Zbl 0207.13501 |
| [14] | M. Taylor, Pseudodifferential operators , Princeton Mathematical Series 34 , (1981) Princeton Univ. Press. · Zbl 0453.47026 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
