A short course in K-theory by Marc Levine

By Marc Levine

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Since π1 (X ˜ 1 , ∗) → π1 (X, ˜ ∗) is N . Let X 1 , X 1 ˜ surjective, we can lift yi , zi to y¯i , z¯i in π1 (X , ∗). Attach a two-cell Dx2 to X by the attaching map p∗ ( i [¯ yi , z¯i ]), forming the space Y . Similarly, 2 for each g ∈ G, attach a two cell Dx,g to g( i [¯ yi , z¯i ]), forming the space 2 2 ˜ ˜ Y . Extend the G action to Y by sending Dx,g to Dx,g g via the identity. This makes Y˜ → Y a covering space with group G. Continue doing this for enough x ∈ N to generate N as a normal subgroup, and denote again by Y˜ → Y the resulting spaces.

9. 2, BQ(MZ ) → BQ(MX (Z)) is a weak equivalence, which proves the theorem. 5. We now have two possible definitions of the K-theory of a commutative ring R, namely Kp (R) and Kp (Spec R). In [15], it is shown that there is a natural isomorphism of these two, even as H-spaces. 2. Mayer-Vietoris. 6. Let X be a scheme, jU : U → X, jV : V → X open subschemes with X = U ∪ V . Let j0 : U ∩ V → U , j1 : U ∩ V → V be the inclusions. Then (BQj ∗ ,BQj ∗ ) BQj ∗ −BQj ∗ 2 1 −−−→ BQ(MU ∩V ) BQ(MX ) −−−−U−−−−V→ BQ(MU ) × BQ(MV ) −−−− is a weak homotopy fiber sequence, giving the Mayer-Vietoris sequence ∂ .

Three fundamental properties of the functor K0 extend in a much stronger fashion to the higher K-groups. To explain this, we first recall a basic notion from algebraic topology. Let f : (X, ∗) → (Y, ∗) be a continuous map of pointed topological spaces. The homotopy fiber of f is the space F ib(f ) consisting of pairs (x, γ), where x is in X, and γ is a path from f (x) to ∗ ∈ Y ; the topology on F ib(f ) is induced from that of X and Y . The base-point on F ib(f ) is (∗X , id∗Y ). We have the map q : (F ib(f ), ∗) → (X, ∗) by sending (x, γ) to x; clearly the paths γ give a canonical homotopy of f ◦ q to the map F ib(f ) → ∗.

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