It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(4)\to K({\mathbb Z}/2,3)$ is 5-connected. Is it known whether $\mathrm{TOP}(4)/\mathrm{PL}(4)$ is not equivalent to $K({\mathbb Z}/2,3)$?