At the bottom of p1, the assumption that ##U^+## lies in the closed upper half-ball will not hold if the point ##p## has zero ##i##th coordinate.
It looks like you are defining a global map ##\phi## and trying to show its restriction to a nbd of a boundary point on the hypersphere is a homeomorphism to a half-ball. I don't think that will work because of equator points wrt the ##i##th dimension. Instead, choose the putative homeomorphism based on the location of the boundary point. A boundary point must have at least one nonzero coordinate. Say the first nonzero coord of the point is the ##i##th coordinate, then define ##\pi_i## to be the projection map that removes the ##i##th coord.
That should enable you to get around the obstacle.
