18.917: Topics in algebraic topology

The course will give an introduction to higher algebraic K-theory following [1] and [2]. To give an idea of what this is about, suppose first that k is a field. The determinant det: GL_{n}(k) → k^{∗} factors through the group homomorphisms

GL_{n}(k) ↪ GL(k) ↠ GL(k)^{ab},

and the induced map GL(k)^{ab} ⟶^{∼} k^{∗}, it turns out, is an isomorphism. If k is any ring, we still have the group homomorphisms above, and therefore, we can define the “determinant” of M ∈ GL_{n}(k) to be the image [M] ∈ GL(k)^{ab}. The abelian group K_{1}(k) = GL(k)^{ab} is the first algebraic K-group or the Whitehead group.

There is no algebraic definition definition of the higher K-groups. Instead, let BGL(k) be the classifying space with fundamental group GL(k) and with all higher homotopy groups trivial. There is a space K(k) which can be thought of as the “abelianization” of BGL(k). The fundamental group of K(k) is the group K_{1}(k) = GL(k)^{ab}, but the construction also introduces higher homotopy groups. These, by definition, are the higher K-groups, K_{q}(k) = π_{q}K(k).

Clearly, the groups K_{q}(k) depend only on the homotopy type of K(k), and it is useful to consider many different homeomorphism types within this homotopy type. One example of a result we will prove is the so-called localization sequence: Let V be discrete valuation ring with field of fractions K and residue field k. Then there is a long-exact sequence of K-groups

⋯ → K_{q}(k) → K_{q}(V) → K_{q}(K) ⟶^{∂} K_{q−1}(k) → …,

which for q = 1 is isomorphic to the sequence

0 → V^{∗} → K^{∗} ⟶^{v} ℤ → 0.

References

[1] D. Quillen, Higher algebraic K-theory I, Algebraic K-theory I: Higher K-theories (Battelle Memorial Inst., Seattle, Washington, 1972), Lecture Notes in Mathematics, vol. 341, Springer-Verlag, 1973.

[2] F. Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology, Lecture Notes in Mathematics, vol. 1126, Springer-Verlag, 1985, pp. 318–419.