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.



H_3 of SL(n,Z) and SL(n,F_p)
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](http://www.math.rutgers.edu/~weibel/Kbook.html).


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](http://www.jstor.org/stable/1971055)), 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).


