HOMOTOPY INVARIANCE OF ALMOST FLAT BETTI NUMBERS (original) (raw)
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L2-index theorems, KK-theory, and connections
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Let M be a compact manifold and D a Dirac type differential operator on M . Let A be a C * -algebra. Given a bundle W (with connection) of A-modules over M , the operator D can be twisted with this bundle. One can then use a trace on A to define numerical indices of this twisted operator. We prove an explicit formula for these indices. Our result does complement the Mishchenko-Fomenko index theorem valid in the same situation. We establish generalizations of these explicit index formulas if the trace is only defined on a dense and holomorphically closed subalgebra B.
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Given a gerbe L, on the holonomy groupoid G of the foliation (M, F), whose pull-back to M is torsion, we construct a Connes Φ-map from the twisted Dupont-Sullivan bicomplex of G to the cyclic complex of the L-projective leafwise smoothing operators on (M, F). Our construction allows to couple the K-theory analytic indices of L-projective leafwise elliptic operators with the twisted cohomology of BG producing scalar higher invariants. Finally by adapting the Bismut-Quillen superconnection approach, we compute these higher twisted indices as integrals over the ambiant manifold of the expected twisted characteristic classes.
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We show that some more results from the literature are particular cases of the so-called "invariance under twisting" for twisted tensor products of algebras, for instance a result of Beattie-Chen-Zhang that implies the Blattner-Montgomery duality theorem. * Research partially supported by the CNCSIS project "Hopf algebras, cyclic homology and monoidal categories", contract nr. 560/2009, CNCSIS code ID−69.
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… -AMERICAN MATHEMATICAL SOCIETY, 1997
Abstract. We give short proofs of the Gromov-Shubin theorem on the homotopy invariance of the Novikov-Shubin invariants and of the Dodziuk theorem on the homotopy invariance of the L2 Betti numbers of the universal covering of a closed manifold in this paper. We show ...
The Čech Centennial: A Conference on Homotopy Theory
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We discuss a conjecture of Gromov and Lawson, later modified by Rosenberg, concerning the existence of positive scalar curvature metrics. It says that a closed spin manifold M of dimension n ≥ 5 has a positive scalar curvature metric if and only if the index of a suitable "Dirac" operator in KO n (C * (π 1 (M))), the real K-theory of the group C *-algebra of the fundamental group of M, vanishes. It is known that the vanishing of the index is necessary for existence of a positive scalar curvature metric on M, but this is known to be a sufficient condition only if π 1 (M) is the trivial group, Z/2, an odd order cyclic group, or one of a fairly small class of torsion-free groups. We note that the groups KO n (C * (π)) are periodic in n with period 8, whereas there is no obvious periodicity in the original geometric problem. This leads us to introduce a "stable" version of the Gromov-Lawson conjecture, which makes the weaker statement that the product of M with enough copies of the "Bott manifold" B has a positive scalar curvature metric if and only if the index of the Dirac operator on M vanishes. (Here B is a simply connected 8-manifold which represents the periodicity element in KO 8 (pt).) We prove the stable Gromov-Lawson conjecture for all spin manifolds with finite fundamental group and for many spin manifolds with infinite fundamental group.
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We investigate how one can twist [Formula: see text]-invariants such as [Formula: see text]-Betti numbers and [Formula: see text]-torsion with finite-dimensional representations. As a special case we assign to the universal covering [Formula: see text] of a finite connected [Formula: see text]-complex [Formula: see text] together with an element [Formula: see text] a [Formula: see text]-twisted [Formula: see text]-torsion function [Formula: see text], provided that the fundamental group of [Formula: see text] is residually finite and [Formula: see text] is [Formula: see text]-acyclic.