The norm of the Euler class

We prove that the norm of the Euler class 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. In the course of the proof, we construct a new cocycle representing and taking only the two values ±2 − n . Furthermore, we establish the uniqueness of a canonical bounded Euler class.