By John Roe

ISBN-10: 0821825593

ISBN-13: 9780821825594

"Coarse geometry" is the learn of metric areas from the asymptotic standpoint: metric areas (such because the integers and the true numbers) which "look an analogous from a very good distance" are thought of to be similar. This e-book develops a cohomology conception applicable to coarse geometry. the speculation is then used to build "higher indices" for elliptic operators on noncompact whole Riemannian manifolds. Such an elliptic operator has an index within the $K$-theory of a undeniable operator algebra clearly linked to the coarse constitution, and this $K$-theory then pairs with the coarse cohomology. the better indices may be calculated in topological phrases because of the paintings of Connes and Moscovici. they could even be interpreted in phrases of the $K$-homology of a great boundary obviously linked to the coarse constitution. purposes to geometry are given, and the e-book concludes with a dialogue of the coarse analog of the Novikov conjecture.

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**Extra info for Coarse Cohomology and Index Theory on Complete Riemannian Manifolds**

**Example text**

3 . 1 2 ) LEMMA: Let X and Y be locally compact paracompact Hausdorff spaces, and let fa: X —+Y be a proper homotopy of maps X —+ Y. Suppose that U is a good cover of Y and that V is a good cover of X which is a refinement of f*U for all t e [0,1]. )«-i/;(V;R). 5]. 2. T h e m a i n t h e o r e m Let M be a space (in the category UBB, as usual). We will prove a theorem relating the coarse cohomology of M to the Cech cohomology of certain open covers of M . A novel feature is t h a t whereas in ordinary Cech cohomology one considers finer and finer covers and passes to a direct limit, here we will consider coarser and coarser covers and pass to an inverse limit.

Suppose in particular that X is a Riemannian manifold of dimension n. Then HXq(M) is equal to R if q = n and 0 otherwise. 44) PROPOSITION: IfT is the fundamental group of a compact acyclic nmanifold, and T is equipped with a word metric, then R [0 {q- n) (q £ n) In particular, a surface group has the same coarse cohomology as R . Surface groups have only one relation, but that one relation makes a big difference, as the following proposition shows. 45) PROPOSITION: Let T be a finitely-generated free group, equipped with a word metric.

Niv,_ 1 . 24 JOHN ROE Clearly r is finitely supported, so it belongs to C | " 1 ( W ; T). Moreover, ST = a;. ,Wq) = J2 £ ( - I ) W A ( V O , A « ) , . . ,/iWr;... ,W f ) Therefore «r(Wb , • • • , W,) = £ < * / ! > ( Wo, • • • , W , ) , and since ey is a partition of unity, this gives the result. • ( 3 . 1 1 ) We will also need a version of the 'homotopy l e m m a' for compactly supported cohomology. Suppose t h a t X and Y are locally compact paracompact Hausdorff spaces, and let / : X —• Y be a proper m a p .

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