TITLE: H_3 of SL(n,Z) and SL(n,F_p)
QUESTION [13 upvotes]: Can anyone tell me what $H_3(SL_n(\mathbb{Z});\mathbb{Z})$ and $H_3(SL_n(\mathbb{F}_p);\mathbb{Z})$ are?  It is easy to find references for $H_1$ and $H_2$, but it turns out that I need $H_3$ as well.  All I care about are the stable values.

REPLY [14 votes]: Summarizing the comments, the stable ($n\geq 3$) values of $H_3(SL_n;\mathbb Z)$ are

$H_3(SL_\infty(\mathbb Z);\mathbb Z) = \mathbb Z/24$
$H_3(SL_\infty(\mathbb F_q);\mathbb Z) = \mathbb Z/(q^2-1)$

and can be found in Weibel's The $K$-book.
Namely, one has 
$$
K_2(R)\xrightarrow{[-1]}K_3(R)\to H_3(E(R))\to0
$$
for any ring (Corollary IV.1.20 there), with $K_2(R)=H_2(E(R);\mathbb Z)$ and $K_3(R)=H_3(St(R);\mathbb Z)$, where $E(R)\subseteq GL(R)$ is the subgroup generated by elementary matrices and $St(R)\twoheadrightarrow E(R)$ is its universal central extension.
For most decent rings, including integers and fields, $E_n=SL_n$; for $R$ the integers, $K_2(\mathbb Z)=H_2(SL_n(\mathbb Z))=\mathbb Z/2$ (Milnor), $K_3(\mathbb Z)=H_3(St_n(\mathbb Z))=\mathbb Z/48$ (Lee & Szczarba 1976), and the map $[-1]$ is nonzero. For $R=\mathbb F_q$, $K_2$ is zero and $K_3$ is $\mathbb Z/(q^2-1)$ (Quillen).