◆ S-ASSEMBLY FIBER

S-CONSTRUCTION vs Q-CONSTRUCTION

Waldhausen's S-construction generalizes Quillen's Q-construction. For an exact category F, there is a natural equivalence K^W(F) ≃ K^Q(F). The S-construction applies more broadly: to Waldhausen categories that are not necessarily exact.

APPROXIMATION THEOREM

Waldhausen's Approximation Theorem: If F: C → D is an exact functor between Waldhausen categories satisfying:

  App1: F reflects weak equivalences

  App2: For every map F(A) → X in D, exists A → B in C with F(B) ~→ X

then F induces a homotopy equivalence K(C) ≃ K(D).

FIBER OF ASSEMBLY

The fiber of the assembly map α measures the "exotic" K-theory: classes in K_*(R[G]) not detected by the homological side. The Nil-groups and UNil-groups live in this fiber.

  fib(α) → H_*(BG; K(R)) → K_*(R[G])