Slant products on the Higson-Roe exact sequence

Abstract

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

arXiv comment82 pages; v2: Minor improvements. To appear in Ann. Inst. Fourier

References in corpus (16)

• Expanders, exact crossed products, and the Baum-Connes conjecture
• A Model for the Universal Space for Proper Actions of a Hyperbolic Group
• Rho-classes, index theory and Stolz' positive scalar curvature sequence
• Dualizing the coarse assembly map
• A descent principle for the Dirac dual Dirac method
• Localization C*-algebras and K-theoretic duality
• Large and small group homology
• Codimension two index obstructions to positive scalar curvature
• Singular spaces, groupoids and metrics of positive scalar curvature
• Positive scalar curvature and product formulas for secondary index invariants
• The surgery exact sequence, K-theory and the signature operator
• Infinite loop spaces and positive scalar curvature in the presence of a fundamental group
• A Geometric Description of Equivariant K-Homology for Proper Actions
• The Baum-Connes conjecture with coefficients for word-hyperbolic groups (after Vincent Lafforgue)
• Positive scalar curvature and low-degree group homology
• Coarse and equivariant co-assembly maps

Cited by in corpus (2)

• Band width estimates via the Dirac operator
• Paschke duality and assembly maps