TITLE: Non-vanishing of the Borel classes in the cohomology of $\operatorname{SL}_n(\mathbb Z)$
QUESTION [9 upvotes]: $\DeclareMathOperator\SL{SL}$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 $\SL_n(\mathbb Z)$ 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?

REPLY [2 votes]: 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 by Jeffrey Giansiracusa, myself, and Bena Tshishiku. In particular, Lee's result is true.