The norm of the Euler class (Q425136)

The norm of an element in the group cohomology \(H^{\bullet} (G,\mathbb R)\) is defined as the infinimum of the sup-norms of all cocycles representing the cohomology class in the bar-resolution. This paper computes the norm of the Euler class. More specifically, let \(n\) be an even integer, and let \({\mathcal E}\) be the Euler class in \(H^n({\mathbf{GL}}_n^+({\mathbb R}),{\mathbb R})\), then the authors show that the norm of \({\mathcal E}\) is \(2^{-n}\). A cocycle representing the Euler class and having sup-norm \(2^{-n}\) has been studied by Sullivan, Smillie, Ivanov and Turaev. Thus this paper proves that this upper bound is sharp. To prove the equality, the paper studies the bounded cohomology \(H^n_{\mathrm b}({\mathbf{GL}}_n^+({\mathbb R}),{\mathbb R})\). First, the authors show that there is a unique bounded Euler class \({\mathcal E}_{\mathrm b}\) whose semi-norm is the same as the norm of \({\mathcal E}\), by proving that the space of \textit{antisymmetric} classes in \(H^n_{\mathrm b}({\mathbf{GL}}_n^+({\mathbb R}),{\mathbb R})\) is one-dimensional. The authors exhibit an explicit, algebraically defined invariant on the space of complete oriented flags in \({\mathbb R}^n\) taking only the two values \(\pm 2^{-n}\). This computes the semi-norm of the bounded Euler class. At the end, the relation of this new cocycle with the simplicial cocycles of Sullivan, Smillie, Ivanov and Turaev is clarified.