Let BTOP and BPL be the classifying spaces of topological/PL-sphere bundles and $TOP/PL$ the homotopy fiber of the map $BPL \to BTOP$. The $TOP/PL$ is a model for a $K(\mathbb{Z}/2\mathbb{Z},3)$ by Kirby and Siebenmann. This identifies a third cohomology class as obstruction to get a PL-structure on a topological sphere bundle.

$\DeclareMathOperator\B{B}\newcommand\TOP{\mathrm{TOP}}\newcommand\PL{\mathrm{PL}}\newcommand\BTOP{{\B}\TOP}\newcommand\BPL{{\B}\PL}$Let $\BTOP$ and $\BPL$ be the classifying spaces of topological/PL-sphere bundles and $\BTOP/\BPL$ the homotopy fiber of the map $\BPL \to \BTOP$. Then $\TOP/\PL$ is a model for a $K(\mathbb{Z}/2\mathbb{Z},3)$ by Kirby and Siebenmann. This identifies a third cohomology class as obstruction to get a PL-structure on a topological sphere bundle.

Let BTOP and BPL be the classifying spaces of topological/PL-sphere bundles and $PL/TOP$ the homotopy fiber of the map $BPL \to BTOP$. The $PL/TOP$ is a model for a $K(\mathbb{Z}/2\mathbb{Z},3)$ by Kirby and Siebenmann. This identifies a third cohomology class as obstruction to get a PL-structure on a topological sphere bundle.

Let BTOP and BPL be the classifying spaces of topological/PL-sphere bundles and $TOP/PL$ the homotopy fiber of the map $BPL \to BTOP$. The $TOP/PL$ is a model for a $K(\mathbb{Z}/2\mathbb{Z},3)$ by Kirby and Siebenmann. This identifies a third cohomology class as obstruction to get a PL-structure on a topological sphere bundle.