  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.
