The norm of the Euler class

TL;DR

This paper determines the exact norm of the Euler class for flat vector bundles, showing the sharpness of a known bound, and constructs a unique cocycle representation with specific properties.

Contribution

It proves the precise norm of the Euler class, constructs a new cocycle with minimal values, and establishes the uniqueness of an antisymmetric representative in bounded cohomology.

Findings

The norm of the Euler class in even dimensions is 2^{-n}.

A new cocycle representing the Euler class takes only the values ±2^{-n}.

The Sullivan–Smillie bound is sharp for the Euler class.

Abstract

We prove that the norm of the Euler class 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 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 E in bounded cohomology.