2 8 Ju l 2 00 5 The coarse geometric Novikov conjecture and uniform convexity
نویسنده
چکیده
The classic Atiyah-Singer index theory of elliptic operators on compact manifolds has been vastly generalized to higher index theories of elliptic operators on noncompact spaces in the framework of noncommutative geometry [5] by Connes-Moscovici for covering spaces [8], Baum-Connes for spaces with proper and cocompact discrete group actions [2], Connes-Skandalis for foliated manifolds [9], and Roe for noncompact complete Riemannian manifolds [33]. These higher index theories have important applications to geometry and topology. In the case of a noncompact complete Riemannian manifold, the coarse geometric Novikov conjecture provides an algorithm to determine when the higher index of an elliptic operator on the noncompact complete Riemannian manifold is nonzero. The purpose of this paper is to prove the coarse geometric Novikov conjecture under a certain mild geometric condition suggested by Gromov [16]. Let Γ be a metric space; let X be a Banach space. A map f : Γ → X is said to be a (coarse) uniform embedding [15] if there exist non-decreasing functions ρ1 and ρ2 from R+ = [0,∞) to R such that (1) ρ1(d(x, y)) ≤ ‖f(x)− f(y)‖ ≤ ρ2(d(x, y)) for all x, y ∈ Γ; (2) limr→+∞ ρi(r) = +∞ for i = 1, 2. ∗Partially supported by NSF and NSFC.
منابع مشابه
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تاریخ انتشار 2005