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Index theory with bounded geometry, the uniformly finite $$\widehat A$$ class, and infinite connected sums. (English) Zbl 1031.58013
Let $$M$$ be an $$n$$-manifold of bounded geometry, i.e. a Riemannian manifold with bounds on its curvature tensor and its derivatives, and on the injectivity radius. A subset $$S$$ of $$M$$ is called uniformly discrete if there is an $$\varepsilon>0$$ such that any two distinct points of $$S$$ are at least $$\varepsilon$$ apart. We say that an $$i$$-chain $$c$$ in $$M$$ is uniformly finite if there is a bound on the diameter of simplices in the support of $$c$$ and for every $$r>0$$ there is an upper bound $$C_r$$ on the sum of the absolute values of the coefficients of the simplices which intesects any $$r$$-ball. We denote these chains by $$C_i^{uf}(M)$$ and the corresponding homologies by $$H_i^{u f}(M)$$. Any $$S$$ as above gives a natural element of $$C_0^{uf}(M)$$ and the corresponding class $$[S]$$ in $$H_0^{u f}(M)$$.
The author proves that if the above $$M$$ is complete and connected, then there is a canonical isomorphism between $$H_0^{uf}(M)$$ and $$H^n_\infty(M)$$. Moreover the $$\ell^\infty$$-cohomology class of the $$\widehat A$$ characteristic class is indepedent of the choice of metrics (within the given bounded distortion class). From these considerations, the authors defines the $$\widehat A^{uf}$$ genus, and proves the following main theorem: If $$M$$ has non-negative scalar curvature (of bounded geometry), then $$\widehat A^{uf}(M)=0$$. Combining this within the work of J. Block and S. Weinberger [J. Am. Math. Soc. 5, 907-918 (1992; Zbl 0780.53031)], the author gives a complete characterization of infinite connected sums of positive scalar curvature.

##### MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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