By Lewis

ISBN-10: 2152164164

ISBN-13: 9782152164168

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**Sample text**

The Bing school in topology created extraordinarily sophisticated techniques to study decomposition spaces. One of the primary questions was to understand the conditions under which X/G is an nmanifold if X is an n-manifold. Even if X/G is not a manifold, it is in many cases a homology manifold, which means that it cannot be distinguished from a manifold by means of local algebraic topology. 9) 17By H∗ (Y, Y \ {y} : Z) = H∗ (Rn , Rn \ {0} : Z) , the lower bound for the Ricci curvature we mean that the Ricci curvature tensor Ric satisfies Ric≥ kg, where g is the Riemannian metric.

See [96], [184], [79, Section 11]. The validity of a Poincar´e inequality carries over to Gromov-Hausdorff limits of metric measure spaces, where a convergence of measures has to be incorporated in the definition. 6. 8 and the fact that every complete Riemannian manifold NONSMOOTH CALCULUS 53 with nonnegative Ricci curvature supports a 1-Poincar´e inequality [41]. See [47] for applications. Some singular spaces that arise as boundaries of Gromov hyperbolic groups support a Poincar´e inequality; in many cases, it is not known whether or not a Poincar´e inequality holds.

7. Tangent spaces. In general, there are many tangent spaces at a given point, depending on the small scales considered. A convenient class of metric spaces, where tangent spaces can be expected, is the class of doubling spaces. A metric space is said to be doubling if there exists a constant C ≥ 1 such that every ball in the space can be covered by at most C balls of half the radius. The doubling condition promotes itself to the following ostensibly stronger requirement: there exist constants C ≥ 1 and α > 0 such that the cardinality of every maximal R-net, 0 < ≤ 12 , in a ball of radius R > 0 does not exceed C −α .

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