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arXiv · 2107.02843

Paschke duality and assembly maps

Ulrich Bunke·Alexander Engel·Markus Land
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Abstract

We construct a natural transformation between two versions of $G$-equivariant $K$-homology with coefficients in a $G$-$C^{*}$-category for a countable discrete group $G$. Its domain is a coarse geometric $K$-homology and its target is the usual analytic $K$-homology. Following classical terminology, we call this transformation the Paschke transformation. We show that under certain finiteness assumptions on a $G$-space $X$, the Paschke transformation is an equivalence on $X$. As an application, we provide a direct comparison of the homotopy theoretic Davis-L\"uck assembly map with Kasparov's analytic assembly map appearing in the Baum-Connes conjecture.

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Ulrich Bunke, Alexander Engel, Markus Land. 2021-07-06. Paschke duality and assembly maps. https://arxiv.org/abs/2107.02843

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