K. Habiro gave a neccesary and sufficient condition for knots to have the same Vassiliev invariants in terms of -moves. In this paper we give another geometric condition in terms of Brunnian local moves. The proof is simple and self-contained.
We calculate the mapping and obtain a generating system of its kernel. As a corollary, bounds on the codimension of fold maps from real projective spaces to Euclidean space are calculated and the rank of a singular bordism group is determined.
The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map.