Cup product in bounded cohomology of negatively curved manifolds

Abstract

Let $M$ be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential $2$-form $\xi \in {\Omega }^2(M)$ defines a bounded cocycle $c_{\xi }\in C_b^2(M)$ by integrating $\xi$ over straightened $2$-simplices. In particular Barge and Ghys proved that, when $M$ is a closed hyperbolic surface, ${\Omega }^2(M)$ injects this way in $H_b^2(M)$ as an infinite dimensional subspace. We show that any class of the form $[c_{\xi }]$, where $\xi$ is an exact differential 2-form, belongs to the radical of the cup product on the graded algebra $H_b^{\bullet }(M)$.

9 pages

Cited by in corpus (1)

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