31 research outputs found

    Band width estimates via the Dirac operator

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    Let MM be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on V=M×[1,1]V = M \times [-1,1] with scalar curvature bounded below by σ>0\sigma > 0, the distance between the boundary components of VV is at most Cn/σC_n/\sqrt{\sigma}, where Cn=(n1)/nCC_n = \sqrt{(n-1)/{n}} \cdot C with C<8(1+2)C < 8(1+\sqrt{2}) being a universal constant. This verifies a conjecture of Gromov for such manifolds. In particular, our result applies to all high-dimensional closed simply connected manifolds MM which do not admit a metric of positive scalar curvature. We also establish a quadratic decay estimate for the scalar curvature of complete metrics on manifolds, such as M×R2M \times \mathbb{R}^2, which contain MM as a codimension two submanifold in a suitable way. Furthermore, we introduce the "KO\mathcal{KO}-width" of a closed manifold and deduce that infinite KO\mathcal{KO}-width is an obstruction to positive scalar curvature.Comment: 24 pages, 2 figures; v2: minor additions and improvements; v3: minor corrections and slightly improved estimates. To appear in J. Differential Geo

    Slant products on the Higson-Roe exact sequence

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    We construct a slant product / ⁣:Sp(X×Y)×K1q(credY)Spq(X)/ \colon \mathrm{S}_p(X \times Y) \times \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X) on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer. The latter is the domain of the co-assembly map μ ⁣:K1(credY)K(Y)\mu^\ast \colon \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{K}^\ast(Y). We obtain such products on the entire Higson--Roe sequence. They imply injectivity results for external product maps. Our results apply to products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the class of manifolds where this method applies, we say that a complete spinc\mathrm{spin}^{\mathrm{c}}-manifold is Higson-essential if its fundamental class is detected by the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. We draw conclusions for positive scalar curvature metrics on product spaces, particularly on non-compact manifolds. We also obtain equivariant versions of our constructions and discuss related problems of exactness and amenability of the stable Higson corona.Comment: 82 pages; v2: Minor improvements. To appear in Ann. Inst. Fourie
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