Odd-degree reduced exact bounded cohomology of torus homeomorphism groups

Determine whether the reduced exact bounded cohomology $EH^d\bigl(Homeo_0(T^2,\mu)\bigr)$ is infinite-dimensional for every odd integer $d>3$.

Background

Theorem 5.1 establishes infinite-dimensional reduced bounded cohomology for the area-preserving homeomorphism group of the torus in every degree at least two, but its reduced exact conclusion for this group is obtained only in even degrees. The torus argument uses a two-point transfer and an infinite-dimensional subspace of degree-two classes with exact images; exactness of the relevant degree-three factor for torus homeomorphisms is not established.

The unresolved question asks whether the missing odd-degree exact-bounded-cohomology conclusion holds in all odd degrees above three, extending the theorem’s even-degree exactness result.