Crossed flux homomorphisms and vanishing theorems for flux groups
  We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the manifold resembles one with a circle action with homologically essential orbits. 
J. Kedra, D. Kotschick, S. Morita