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 codimension two submanifold of $M$, then one can conclude that the $n$-th bounded cohomology of the fundamental group of $M\setminus S$ is non-zero. While in this note this approach is only used for degree $4$. There is no reason to expect that this approach and its generalizations is not suitable to prove the non-vanishing of higher degrees or the bounded cohomology of different groups as well.