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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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