31 research outputs found

### Band width estimates via the Dirac operator

Let $M$ 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 $V = M \times [-1,1]$ with scalar curvature bounded below by $\sigma
> 0$, the distance between the boundary components of $V$ is at most
$C_n/\sqrt{\sigma}$, where $C_n = \sqrt{(n-1)/{n}} \cdot C$ with $C <
8(1+\sqrt{2})$ 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 $M$ 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 $M \times \mathbb{R}^2$,
which contain $M$ as a codimension two submanifold in a suitable way.
Furthermore, we introduce the "$\mathcal{KO}$-width" of a closed manifold and
deduce that infinite $\mathcal{KO}$-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

We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \times
\mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ 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 $\mu^\ast \colon \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \to
\mathrm{K}^\ast(Y)$. 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
$\mathrm{spin}^{\mathrm{c}}$-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