Slant products on the Higson–Roe exact sequence

Abstract

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.