An assembly map in index theory is a natural comparison map from topological K-homology data to the topological K-theory of an associated operator algebra (Higson and Roe 2000). It sends geometric cycles to analytic index classes. The domain, coefficients, support convention, and completion of the algebra are part of the definition of each version.
For a torsion-free discrete group 
,
reduced group assembly has the form
 |
where 
is a classifying space, the domain uses compact
supports, and 
is the reduced group C*-algebra.
The Baum-Connes conjecture asks for an
isomorphism. Coarse assembly instead takes the direct limit of locally finite K-homology
of Rips complexes of a metric
space, with a Roe algebra as target. The coarse
Novikov conjecture concerns rational injectivity of this latter map.
