  Equivariant homotopy methods developed over the last 20 years lead to recent
breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in
K- and L-theories. An important consequence of these algebraic conjectures is
the topological rigidity of compact aspherical manifolds. Our goal is to strip
the basic idea to the core and follow the evolution over time in order to
explain the advantages of the flexible state that exists today. We end with an
outline of the proof of the Borel conjecture in algebraic K-theory for groups
of finite asymptotic dimension. We also discuss the relation of these methods
to the recent work on the Farrell-Jones conjecture.
