Algebraic K-theory

Let $A$ be a ring with unit. Define a discrete group $\GL(A)$ as a direct limit of the groups $\GL_r(A)$ with respect to the maps

$$\GL_r(A)\hookrightarrow \GL_{r+1}(A), \quad \al\mapsto \twobytwo{\al}{0}{0}{1}.$$

There is a classifying space $\B \GL(A)$ with
$$\begin{align*} \pi_1(\B \GL(A)) &= \GL(A),\\ \pi_n(\B \GL(A)) &= 0, \quad n\neq 1. \end{align*}$$
We can apply the Quillen's plus construction to obtain a space $\B \GL(A)^+$ with the following three properties

1. the fundamental group is an abelianization of $\GL(A)$,
   
   $$\pi_1(\B \GL(A)^+)=\GL(A)/[\GL(A), \GL(A)],$$
   
   

2. there is an isomorphism on homology $\rH_i(\B \GL(A))\isom \rH_i(\B \GL(A)^+)$,
3. there is an H-space structure on $\B \GL(A)^+$.

Thus $\rH_*(\B\GL(A)^+)$ is commutative, cocommutative (and connected) Hopf algebra.

Prior to this definition there were defined $\rK_1$, $\rK_2$, $\rK_3$. We will describe these earlier definitions.

The $\rK_1$ group of a ring $A$ was defined as an abelianization of $\GL(A)$,

$$\rK_1(A)=\GL(A)/[\GL(A), \GL(A)].$$

For example if $A=F$ is a field, then $\rK_1(F)=F^{\times}$, the group of invertible elements in $F$. The determinant map $\det\:\GL(F)\to F^{\times}$ can be generalized to noncommutative rings by the map $\GL(A)\to \rK_1(A)$.

Denote by $\El(A)$ the group generated by elementary matrices $e_{ij}^a$, where each $e_{ij}^a$ is an identity matrix plus the matrix with only one nonzero entry equal $a$ in $i$-th row and $j$-th column. Then

$$[\GL(A), \GL(A)]=\El(A).$$

The elementary matrices $e_{ij}^a$ satisfy the following relations

$$\begin{cases} e_{ij}^a e_{ij}^b &= e_{ij}^{a+b},\\ e_{ij}^a... ...e_{ij}^a e_{jk}^b &= e_{jk}^b e_{ik}^{ab} e_{ij}^a. \end{cases}$$ (4)

The group $\El(A)$ can be presented using generators $e_{ij}^a$ which satisfy the relations () above plus some relations which depend on $A$. Define the Steinberg group $\St(A)$ of $A$ as the group with the set of generators $\{x^a_{ij}\}$ with the relations (). There is an epimorphism $\St(A)\epi \El(A)$ and we define $\rK_2(A)$ as the kernel of this map. Then $\rK_2(A)$ is abelian, and the sequence

$$\rK_2(A)\mono\St(A)\epi \El(A)$$

is a central extension.







Summarizing earlier results we have
$$\begin{align*} \rH_1(\GL(A)) &= \rK_1(A), \\ \rH_2(\El(A)) &= \rK_2(A), \\ \rH_3(\St(A)) &= \rK_3(A). \end{align*}$$
Let us look once more at the relations for Steinberg group (). We can label the edges of a Stasheff polytope of dimension 2 as follows

$$\xymatrix{ & \bullet \ar[ld]_{e_{jk}^b} \ar[rd]^{e_{ij}^a} &... ...{e_{jk}^b} \\ \bullet \ar[rd]_{e_{ij}^a} & &\\ & \bullet & }$$

to encode the relation $e_{ij}^a e_{jk}^b = e_{jk}^b e_{ik}^{ab} e_{ij}^a$. There is a way to put labels on the Stasheff polytope of dimension 3 in the coherent way. It can be generalized to higher dimensions.

Using Hurewicz homomorphism

$$h\:\pi_*(\B \GL(A)^+)\to \rH_*(\B \GL(A)^+; \bQ)$$

we can define a map

$$\xymatrix{ \rK_*(A)\ox \bQ \ar[r]^{h} \ar[rd] & \rH_*(\B \GL(A); \bQ) \ar@{-->}[d] \\ & \HC^-_*(A) }$$