1,721,166 research outputs found
Manifolds with Boundary and of Bounded Geometry
No full textFor non–compact manifolds with boundary we prove that bounded geometry defined by coordinate–free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geometry and we study the change of geodesic coordinate maps
^2ehBindex theorem for elliptic differential boundary problems
No full textSuppose M is a compact manifold with boundary ∂M. Let M be a normal covering of M. Suppose (A,T) is an elliptic differential boundary value problem on M with lift (Ã,T) to M. Then the von Neumann dimension of kernel and cokernel of this lift are defined. The main result of this paper is: These numbers are finite, and their difference, by definition the von Neumann index of (Ã,T), equals the index of (A,T). In this way, we extend the classical L2-index theorem of Atiyah to elliptic differential boundary value problems
Real versus complex K-theory using Kasparov's bivariant KK-theory
No full textIn this paper, we use the KK–theory of Kasparov to prove exactness of sequences relating the K–theory of a real C∗–algebra and of its complexification (generalizing results of Boersema). We use this to relate the real version of the Baum-Connes conjecture for a discrete group to its complex counterpart. In particular, the complex Baum–Connes assembly map is an isomorphism if and only if the real one is, thus reproving a result of Baum and Karoubi. After inverting 2, the same is true for the injectivity or surjectivity part alone
Erratum to Integrality of L2-Betti numbers
No full textA gap has been found in the proof of Proposition 3 of [3]. Consequently, this result and the consequences drawn in the paper have to be considered as conjectures. In particular, it is not proved that the Atiyah conjecture is stable under extensions with free quotient groups and under free products with free groups
Integrality of -Betti numbers
No full textThe Atiyah conjecture predicts that the L2-Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We establish the Atiyah conjecture, under the condition that it holds for G and that H⊲G is a normal subgroup, for amalgamated free products G∗H(H⋊F). Here F is a free group and H⋊F is an arbitrary semi-direct product. This includes free products G F and semi-direct products G⋊F. We also show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new (bigger) class of groups for which the Atiyah conjecture holds, which contains all free groups and in particular is closed under taking subgroups, direct sums, free products, extensions with torsion-free elementary amenable quotient or with free quotient, and under certain direct and inverse limits
A counterexample to the (unstable) Gromov–Lawson–Rosenberg conjecture
No full textAbstractDoing surgery on the 5-torus, we construct a five-dimensional closed spin-manifold M with π1(M)≅Z4×Z/3, so that the index invariant in the KO-theory of the reduced C∗-algebra of π1(M) is zero. Then we use the theory of minimal surfaces of Schoen/Yau to show that this manifold cannot carry a metric of positive scalar curvature. The existence of such a metric is predicted by the (unstable) Gromov–Lawson–Rosenberg conjecture
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