The norm of 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.

19 pages

References in corpus (2)

• The simplicial volume of closed manifolds covered by H^2 x H^2
• On the bounded cohomology of semi-simple groups, S-arithmetic groups and products

Cited by in corpus (1)

• Bounded cohomology via partial differential equations, I