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 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math20_1.jpg"/> be a family of subgroups of G, i.e., a collection of subgroups closed under conjugation and passing to subgroups. Let O r ( G ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math20_2.jpg"/> be the orbit category and O r F ( G ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math20_3.jpg"/> be the full subcategory consisting of objects G/H satisfying H ∈ F https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math20_4.jpg"/> . Consider a covariant functor E G : O r ( G ) → S p e c t r a https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math20_5.jpg"/> to the category of spectra. We get from the inclusion O r F ( G ) → O r ( G ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math20_6.jpg"/> and the fact that G/G is a terminal object in O r ( G ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math20_7.jpg"/> a map hocolim O r F ( G ) E G | O r F ( G ) → hocolim O r ( G ) E G = E G ( G / G ) . https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/math20_1_1.jpg"/> It is called assembly map since we are trying to assemble the values of E G on homogeneous spaces G/H for H ∈ F https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math20_8.jpg"/> to get E(G/G).
