It is known that $H^k(SL(n,\mathbb{Z}))$ is independent of $n$ for $n \gg k$, so we can define a stable cohomology ring $$V = \text{lim}_{n \rightarrow \infty} H^{\ast}(SL(n,\mathbb{Z});\mathbb{R}).$$ Using analytic tools, Borel proved that $V$ is an exterior algebra generated in degrees $3, 5, 7, \ldots$. Is there proof of this using more traditional algebraic topology tools?
