Non vanishing of the fourth bounded cohomology of free groups and codimension 2 subspaces

In this note we prove that the fouth bounded cohomology of non-abelian free groups with trivial real coefficients is non-zero. In order to prove this, we establish a splitting argument whose simplest form is as follows: Let $M$ denote an $n$-manifold of non-zero simplicial volume and $S$ a c