Abstract
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. In the course of the proof, we construct a new cocycle representing \({\mathcal {E}}\) and taking only the two values ±2−n. Furthermore, we establish the uniqueness of a canonical bounded Euler class.
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M. Bucher acknowledges support by the Swiss National Science Foundation (Grants PZ00P2-126410 and PP00P2-128309/1). N. Monod acknowledges support in part by the Swiss National Science Foundation (Grant CRSI22-130435) and the European Research Council.