Coarse cohomology for families (Q5954275)

The authors introduce the definition of a parametrized family of metric spaces, the main example of which is a Hausdorff foliation groupoid. They define the coarse cohomology of such a metric family in the spirit of \textit{J. Roe}'s work [Index theory, coarse geometry, and topology of manifolds, Reg. Conf. Ser. Math. 90 (1996; Zbl 0853.58003)], utilizing a fiberwise coarsening map. The coarse cohomology is shown to be a fiberwise coarse invariant --- that is, dependent only on the coarse type of the fibers. For the case of foliations on compact manifolds with Hausdorff graphs, the researchers define three more natural coarse theories, the de Rham, Čech, and Alexander-Spanier theories, and they show that all three are isomorphic to the originally defined coarse cohomology. If the graph of the foliation is a fiber bundle over the manifold, then there is a spectral sequence (similar to the spectral sequence of a fiber bundle) that converges to the coarse cohomology of the metric family of leaves. It is shown that foliations of compact manifolds that are leafwise homotopy equivalent have isomorphic coarse cohomology. The authors also show that the coarse cohomology maps to the usual cohomology with compact supports of the graph of the foliation; under certain conditions investigated in this paper such as uniform contractibility or rescalability, this map is an isomorphism. This paper is well-written and contains an assortment of nice examples.