where last factor is not simply connected: Geometrically, it is seen (for visualization purposes, S 3 may be replaced by S 2 ) that a path from u to − u in S U ( 2 ) ≅ ≅ S 3 is a loop in S 3 / Z 2 since u and − u are antipodal points, and that it is not contractible to a point. But a path from u to − u, thence to u again, a loop in S 3 and a double loop (considering p (ue) = p (− ue), where p : SL ( 2 , C ) → → SO ( 3 ; 1 ) is the covering map) in S 3 / Z 2 that is contractible to a point (continuously move away from − u "upstairs" in S 3 and shrink the path there to the point u). Thus π 1 (SO(3; 1)) is a group with two equivalence classes of loops as its elements, or put more simply, SO(3; 1) is doubly connected.