# Non-vanishing of the Borel classes in the cohomology of $\operatorname{SL}_n(\mathbb Z)$

$$\DeclareMathOperator\SL{SL}$$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?

• There is a paper by Jens Franke on " a topological model for some summand of Eisenstein cohomology" (see the math review ams.org/mathscinet-getitem?mr=2402680 ) . The section 7.5 deals with the case of $SL_n(\mathbb Z)$ ; Franke's result is very general and it is an exercise to read the special case of SL(n) from his paper. (I hope the following link to the chapter works:) rdcu.be/xQhw Oct 29 '17 at 3:19