It is well known that if the Borel fibration of a compact abelian connected group action on a compact space is totally nonhomologous to zero, then the action must have fixed points. But this is usually not true. In this article, we will give an example that are both TNHZ and the action has no fixed point. Further, we will examine paracompact spaces with TNHZ for compact abelian connected group actions without fixed points, and provide certain necessary conditions for the existence of such actions.