36. The Hauptvermutung has been shown to be true for finite simplicial complexes (which are more general than manifolds) of dimension less than three and to be false for such complexes of dimension greater than five. It is correct for manifolds of dimension less than or equal to three and an open problem for manifolds of dimension four or greater. See John Milnor, Annals of Math., (2), 74, 1961, 575-90.