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