Abstract

The quickest and probably for a homotopy theorist most convenient approach to assembly maps is via homotopy colimits as explained in Subsection 20.7.3. Let F be a family of subgroups of G, i.e., a collection of subgroups closed under conjugation and passing to subgroups. Let O r ( G ) be the orbit category and O r F ( G ) be the full subcategory consisting of objects G/H satisfying H ∈ F . Consider a covariant functor E G : O r ( G ) → S p e c t r a to the category of spectra. We get from the inclusion O r F ( G ) → O r ( G ) and the fact that G/G is a terminal object in O r ( G ) a map 20.1.1 hocolim O r F ( G ) E G | O r F ( G ) → hocolim O r ( G ) E G = E G ( G / G ) . It is called assembly map since we are trying to assemble the values of E ^G on homogeneous spaces G/H for H ∈ F to get E(G/G).