On the computation of coarse cohomology

The purpose of this article is to relate coarse cohomology of metric spaceswith a more computable cohomology. We introduce a notion of boundedly supportedcohomology and prove that coarse cohomology of many spaces are isomorphic tothe boundedly supported cohomology. Boundedly supported cohomology coincideswith compactly supported Alexander--Spanier cohomology if the space is properand contractible. Our work generalizes an earlier result of Roe which says thatthe coarse cohomology is isomorphic to the compactly supportedAlexander-Spanier cohomology if the space is uniformly contractible. As anapplication of our main theorem, we obtain that coarse cohomology of thecomplement can be computed in terms of Alexander-Spanier cohomology for manyspaces.