$\DeclareMathOperator\SL{SL}$The stable real cohomology of $\SL\_n(\mathbb Z)$ was computed by Borel: it is given by $\mathbb R[z\_i\mid i=5,9,13,\dotsc]$ with $z\_i$ in degree $i$. One may wonder whether the pull back of the stable class $z\_i$ on $\SL(\mathbb Z)$ to $\SL\_n (\mathbb Z)$ for some finite $n$ is non-zero. That is, whether $\iota\_n^\* z\_i = 0 \in H^i(\SL\_n(\mathbb Z),\mathbb R)$ for $\iota\_n \colon \SL\_n(\mathbb Z) \to \SL(\mathbb Z)$.


Borel's proof also gives a real cohomological stability result and hence gives a range of $n$'s for which this is true, but in 1978 Ronnie Lee announced a proof in [On unstable cohomology classes of $\SL\_n(\mathbb Z)$](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC411179/) that $z\_i$ is non-zero as long as $i<2n-3$, a much larger range of $n$'s than what can be deduced from Borel's proof. His paper says



> 
> Because the proofs of Theorems 1 and 2 require careful geometric
> construction, it is planned to present the detailed proof later, and
> we will indicate here some of the ideas involved in the case when $n$ is
> odd.
> 
> 
> 


but as far as I know no detailed proof has appeared. My questions are: Has any proof of Ronnie Lee's theorem appeared in the literature? If not, what is the best known range for non-vanishing of the stable Borel classes?



Non-vanishing of the Borel classes in the cohomology of $\operatorname{SL}_n(\mathbb Z)$
You can indeed read this off from the work of Franke, as was done in Section 4.3 of [Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem](https://doi.org/10.2140/tunis.2021.3.75) by Jeffrey Giansiracusa, myself, and Bena Tshishiku. In particular, Lee's result is true.


