Connectedness of the boundary in the $\mathrm{AdS}$/$\mathrm{CFT}$ correspondence

Let M be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary TV of positive scalar curvature.We show that under these conditions, jy n (M;Z) = 0 and in particular N must be connected.These results resolve some puzzles concerning the AdS/CFT correspondence.