Let Formula: see text be a smooth compact oriented connected manifold, and Formula: see text the group of homeomorphisms of Formula: see text supported away from Formula: see text which preserve a Borel probability measure Formula: see text induced by a volume form on Formula: see text, and are isotopic to the identity. In this paper, we identify those Gambaudo-Ghys and Polterovich quasimorphisms Formula: see text which extend Formula: see text-continuously to Formula: see text as quasimorphisms, and to Formula: see text as group cochains whose differentials are semi-bounded cocycles. We present several applications of this result which include unboundedness of certain bi-invariant metric on the commutator subgroup of Formula: see text, and conditions under which a homeomorphism in Formula: see text is undistorted.
Brandenbursky et al. (Fri,) studied this question.