Let $R=\mathbb{Q}[e^{2\pi i /3}]$. Does $H\_3(GL(R))$ have $\mathbb{Z}$-rank $1$? 


If so, what is the index of the map:
$$
\mathbb{Z}\cong K\_3(R)/{\rm Torsion} \to H\_3(GL(R))/{\rm Torsion}\cong \mathbb{Z}?
$$



K-theory of number field
Let $F = \mathbb{Q}(\zeta\_3)$ and $R = \mathbb{Z}[\zeta\_3]$ with a third root of unity $\zeta\_3$. Then the Hurewicz homomorphism 
$$h\_3: K\_3(R) \to H\_3(GL(R); \mathbb{Z})$$ 
induces an isomorphism on the torsion-quotients und the index of the map is $1$. This can be seen as follows: 


1) rank $K\_3(R) = r\_2 = 1$, the number of pairs of complex embeddings of $F$ (see [W, Theorem 1.5]) 


2) Since $R$ is euclidian, $SL(R) = E(R)$, the subgroup generated by elementary matrices. 


3) According to [A, after Cor. 5.1] the Hurewicz homomorphism 
$$h\_n: K\_n(R) = \pi\_nBGL(R)^+ \cong \pi\_nBE(R)^+ \to H\_n(E(R);\mathbb{Z}) = H\_n(SL\_n(R); \mathbb{Z})$$
is surjective for $n=3$ with torsion kernel. 


4) It follows that the ranks of $K\_3(R)$ and $H\_3(SL(R); \mathbb{Z})$ are equal and by 1) they are equal $1$. 


5) $GL(R) \cong SL(R) \times R^\times = SL(R) \times \mathbb{Z} /6 \mathbb{Z}$.


6) From Kunneth formula it follows that $SL(R) \hookrightarrow GL(R)$ induces a monomorphisms 
$$H\_3(SL(R),\mathbb{Z}) \hookrightarrow H\_3(GL(R); \mathbb{Z})$$ 
with torsion cokernel. 


7) It follows from 6) and 3) that the Hurewicz homomorphism 
$$\bar{h}\_3: \pi\_3(R)/(\text{torsion}) \to H\_3GL(R); \mathbb{Z})/(\text{torsion})$$ 
is an isomorphism and the index that was asked for equals $1$. 


  

**Remarks:** 


a) An explicit description of $K\_3(F)$ is given by [W, Theorem 1.7]


b) Note that $K\_n(F) = K\_n(\mathcal{O})$ for $n$ odd (see [W, Theorem 1.6]) 


  

**References:** 


[A] Arlettaz, [The exponent of the homotopy groups of Moore spectra and the stable Hurewicz homomorphism](http://www.math.uiuc.edu/K-theory/0084/StableHu.pdf)


[W] Weibel, [Albebraic K-Theory of Rings of Integers in Local and Global Fields](http://www.math.illinois.edu/K-theory/0691/KZsurvey.pdf)


  

**Edit:** To complete the story: The Hurewicz map 
$$h\_3: K\_3(F) \to H\_3(SL(F);\mathbb{Z})$$
also induces an isomorphism on the torsion-quoients and is of index $1$ there. 


For, from 1) and b) we know that the torsion free part of $K\_3(F)$ has rank $1$ and 3) is also true with $F$ in place of $R$. Since $SL(F) = E(F)$ (this holds for every local ring) the assertin follows from 3). 




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That $H\_3(GL(F);\mathbb{Z})$ is no good choice for approximating $K\_3(F)$ was already discussed in the comments above. 


