Assembly map

In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.

Assembly maps for algebraic K-theory and L-theory play a central role in the topology of high-dimensional manifolds, since their homotopy fibers have a direct geometric interpretation. Equivariant assembly maps are used to formulate the Farrell–Jones conjectures in K- and L-theory.

It is a classical result that for any generalized homology theory ${\displaystyle h_{*}}$ on the category of topological spaces (assumed to be homotopy equivalent to CW-complexes), there is a spectrum ${\displaystyle E}$ such that

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