The norm of the Euler class

We prove that the norm of the Euler class $\mathcal{E}$ for flat vector bundles is $2^{-n}$ (in even dimension $n$, since it vanishes in odd dimension). This shows that the Sullivan-Smillie bound considered by Gromov and Ivanov-Turaev is sharp. We construct a new cocycle representing $\mathcal{E}$ and taking only the two values $\pm 2^{-n}$; a null-set obstruction prevents any cocycle from existing on the projective space. We establish the uniqueness of an antisymmetric representative for $\mathcal{E}$ in bounded cohomology.