Slant products on the Higson–Roe exact sequence

Engel, Alexander; Wulff, Christopher; Zeidler, Rudolf

Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 913-1021.

Nous construisons un slant-produit $/:{\mathrm{S}}_p(X\times Y)\times {\mathrm{K}}_{1-q}\left({𝔠}^{\mathrm{red}}Y\right)\to {\mathrm{S}}_{p-q}\left(X\right)$ sur le groupe structural analytique de Higson et de Roe et la K-théorie de la « stable Higson corona » d’Emerson et de Meyer. Cette dernière est le domaine de définition de l’application de coassemblage ${\mu }^{*}:{\mathrm{K}}_{1-*}\left({𝔠}^{\mathrm{red}}Y\right)\to {\mathrm{K}}^{*}\left(Y\right)$. Nous obtenons ces produits sur toute la suite exacte de Higson–Roe. Ils impliquent que certaines applications produits extérieurs sont injectives. Nos résultats s’appliquent aux produits avec des variétés asphériques dont les groupes fondamentaux se plongent de manière coarse dans un espace de Hilbert. Nous disons qu’une ${\mathrm{spin}}^{\mathrm{c}}$-variété complète est « Higson-essential » si sa classe fondamentale est détectée par l’application de coassemblage. Nous prouvons que les variétés qui sont hyper-euclidiennes coarse sont « Higson-essential » . Nous déduisons des résultats pour des métriques à courbure scalaire positive sur les espaces produits, en particulier sur les espaces non-compacts. En outre, nous donnons des variantes équivariantes de nos constructions et nous discutons l’exactitude et la moyennabilité de la « stable Higson corona » .

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

Classification : 58J22, 19K33, 46L80, 51F30
Keywords: Analytic structure group, K-homology, slant products, assembly maps, exact groups, Higson corona, Novikov conjecture, positive scalar curvature
Mot clés : groupe structural analytique, K-homologie, slant-produits, applications d’assemblage, groupes exacts, Higson corona, conjecture de Novikov, courbure scalaire positive

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