# Anders Björner and Frank H. Lutz. Simplicial manifolds, # bistellar flips and a 16-vertex triangulation # of the Poincaré homology 3-sphere. # Exp. Math. 9, 275-289 (2000). # Anders Björner and Frank H. Lutz. # A 16-vertex triangulation of the Poincaré homology 3-sphere # and non-PL spheres with few vertices. # Electronic Geometry Models, No. 2003.04.001 (2003). #--------------------------------- # Non-PL 5-sphere with 18 vertices # and f-vector f=(18,139,503,904,783,261) non_PL_5_sphere=[ [ 1, 2, 4, 5, 15, 17 ], [ 1, 2, 4, 5, 15, 18 ], [ 1, 2, 4, 5, 16, 17 ], [ 1, 2, 4, 5, 16, 18 ], [ 1, 2, 4, 7, 15, 17 ], [ 1, 2, 4, 7, 15, 18 ], [ 1, 2, 4, 7, 16, 17 ], [ 1, 2, 4, 7, 16, 18 ], [ 1, 2, 5, 13, 15, 17 ], [ 1, 2, 5, 13, 15, 18 ], [ 1, 2, 5, 13, 16, 17 ], [ 1, 2, 5, 13, 16, 18 ], [ 1, 2, 7, 13, 15, 17 ], [ 1, 2, 7, 13, 15, 18 ], [ 1, 2, 7, 13, 16, 17 ], [ 1, 2, 7, 13, 16, 18 ], [ 1, 3, 7, 13, 15, 17 ], [ 1, 3, 7, 13, 15, 18 ], [ 1, 3, 7, 13, 16, 17 ], [ 1, 3, 7, 13, 16, 18 ], [ 1, 3, 7, 15, 16, 17 ], [ 1, 3, 7, 15, 16, 18 ], [ 1, 3, 12, 15, 16, 17 ], [ 1, 3, 12, 15, 16, 18 ], [ 1, 3, 12, 15, 17, 18 ], [ 1, 3, 12, 16, 17, 18 ], [ 1, 3, 13, 15, 17, 18 ], [ 1, 3, 13, 16, 17, 18 ], [ 1, 4, 5, 11, 15, 17 ], [ 1, 4, 5, 11, 15, 18 ], [ 1, 4, 5, 11, 16, 17 ], [ 1, 4, 5, 11, 16, 18 ], [ 1, 4, 7, 9, 15, 17 ], [ 1, 4, 7, 9, 15, 18 ], [ 1, 4, 7, 9, 16, 17 ], [ 1, 4, 7, 9, 16, 18 ], [ 1, 4, 9, 14, 15, 17 ], [ 1, 4, 9, 14, 15, 18 ], [ 1, 4, 9, 14, 16, 17 ], [ 1, 4, 9, 14, 16, 18 ], [ 1, 4, 11, 14, 15, 17 ], [ 1, 4, 11, 14, 15, 18 ], [ 1, 4, 11, 14, 16, 17 ], [ 1, 4, 11, 14, 16, 18 ], [ 1, 5, 11, 15, 17, 18 ], [ 1, 5, 11, 16, 17, 18 ], [ 1, 5, 13, 15, 17, 18 ], [ 1, 5, 13, 16, 17, 18 ], [ 1, 6, 9, 14, 15, 17 ], [ 1, 6, 9, 14, 15, 18 ], [ 1, 6, 9, 14, 16, 17 ], [ 1, 6, 9, 14, 16, 18 ], [ 1, 6, 9, 15, 16, 17 ], [ 1, 6, 9, 15, 16, 18 ], [ 1, 6, 10, 12, 15, 17 ], [ 1, 6, 10, 12, 15, 18 ], [ 1, 6, 10, 12, 16, 17 ], [ 1, 6, 10, 12, 16, 18 ], [ 1, 6, 10, 14, 15, 17 ], [ 1, 6, 10, 14, 15, 18 ], [ 1, 6, 10, 14, 16, 17 ], [ 1, 6, 10, 14, 16, 18 ], [ 1, 6, 12, 15, 16, 17 ], [ 1, 6, 12, 15, 16, 18 ], [ 1, 7, 9, 15, 16, 17 ], [ 1, 7, 9, 15, 16, 18 ], [ 1, 10, 11, 14, 15, 17 ], [ 1, 10, 11, 14, 15, 18 ], [ 1, 10, 11, 14, 16, 17 ], [ 1, 10, 11, 14, 16, 18 ], [ 1, 10, 11, 15, 17, 18 ], [ 1, 10, 11, 16, 17, 18 ], [ 1, 10, 12, 15, 17, 18 ], [ 1, 10, 12, 16, 17, 18 ], [ 2, 3, 9, 11, 15, 17 ], [ 2, 3, 9, 11, 15, 18 ], [ 2, 3, 9, 11, 16, 17 ], [ 2, 3, 9, 11, 16, 18 ], [ 2, 3, 9, 15, 17, 18 ], [ 2, 3, 9, 16, 17, 18 ], [ 2, 3, 11, 14, 15, 17 ], [ 2, 3, 11, 14, 15, 18 ], [ 2, 3, 11, 14, 16, 17 ], [ 2, 3, 11, 14, 16, 18 ], [ 2, 3, 12, 14, 15, 17 ], [ 2, 3, 12, 14, 15, 18 ], [ 2, 3, 12, 14, 16, 17 ], [ 2, 3, 12, 14, 16, 18 ], [ 2, 3, 12, 15, 17, 18 ], [ 2, 3, 12, 16, 17, 18 ], [ 2, 4, 5, 6, 15, 17 ], [ 2, 4, 5, 6, 15, 18 ], [ 2, 4, 5, 6, 16, 17 ], [ 2, 4, 5, 6, 16, 18 ], [ 2, 4, 6, 15, 17, 18 ], [ 2, 4, 6, 16, 17, 18 ], [ 2, 4, 7, 12, 15, 17 ], [ 2, 4, 7, 12, 15, 18 ], [ 2, 4, 7, 12, 16, 17 ], [ 2, 4, 7, 12, 16, 18 ], [ 2, 4, 12, 15, 17, 18 ], [ 2, 4, 12, 16, 17, 18 ], [ 2, 5, 6, 15, 16, 17 ], [ 2, 5, 6, 15, 16, 18 ], [ 2, 5, 13, 15, 16, 17 ], [ 2, 5, 13, 15, 16, 18 ], [ 2, 6, 9, 15, 16, 17 ], [ 2, 6, 9, 15, 16, 18 ], [ 2, 6, 9, 15, 17, 18 ], [ 2, 6, 9, 16, 17, 18 ], [ 2, 7, 10, 13, 15, 17 ], [ 2, 7, 10, 13, 15, 18 ], [ 2, 7, 10, 13, 16, 17 ], [ 2, 7, 10, 13, 16, 18 ], [ 2, 7, 10, 14, 15, 17 ], [ 2, 7, 10, 14, 15, 18 ], [ 2, 7, 10, 14, 16, 17 ], [ 2, 7, 10, 14, 16, 18 ], [ 2, 7, 12, 14, 15, 17 ], [ 2, 7, 12, 14, 15, 18 ], [ 2, 7, 12, 14, 16, 17 ], [ 2, 7, 12, 14, 16, 18 ], [ 2, 9, 11, 15, 16, 17 ], [ 2, 9, 11, 15, 16, 18 ], [ 2, 10, 11, 14, 15, 17 ], [ 2, 10, 11, 14, 15, 18 ], [ 2, 10, 11, 14, 16, 17 ], [ 2, 10, 11, 14, 16, 18 ], [ 2, 10, 11, 15, 16, 17 ], [ 2, 10, 11, 15, 16, 18 ], [ 2, 10, 13, 15, 16, 17 ], [ 2, 10, 13, 15, 16, 18 ], [ 3, 4, 5, 6, 15, 17 ], [ 3, 4, 5, 6, 15, 18 ], [ 3, 4, 5, 6, 16, 17 ], [ 3, 4, 5, 6, 16, 18 ], [ 3, 4, 5, 11, 15, 17 ], [ 3, 4, 5, 11, 15, 18 ], [ 3, 4, 5, 11, 16, 17 ], [ 3, 4, 5, 11, 16, 18 ], [ 3, 4, 6, 8, 15, 17 ], [ 3, 4, 6, 8, 15, 18 ], [ 3, 4, 6, 8, 16, 17 ], [ 3, 4, 6, 8, 16, 18 ], [ 3, 4, 8, 15, 16, 17 ], [ 3, 4, 8, 15, 16, 18 ], [ 3, 4, 11, 14, 15, 17 ], [ 3, 4, 11, 14, 15, 18 ], [ 3, 4, 11, 14, 16, 17 ], [ 3, 4, 11, 14, 16, 18 ], [ 3, 4, 14, 15, 16, 17 ], [ 3, 4, 14, 15, 16, 18 ], [ 3, 5, 6, 10, 15, 17 ], [ 3, 5, 6, 10, 15, 18 ], [ 3, 5, 6, 10, 16, 17 ], [ 3, 5, 6, 10, 16, 18 ], [ 3, 5, 9, 10, 15, 17 ], [ 3, 5, 9, 10, 15, 18 ], [ 3, 5, 9, 10, 16, 17 ], [ 3, 5, 9, 10, 16, 18 ], [ 3, 5, 9, 11, 15, 17 ], [ 3, 5, 9, 11, 15, 18 ], [ 3, 5, 9, 11, 16, 17 ], [ 3, 5, 9, 11, 16, 18 ], [ 3, 6, 7, 8, 15, 17 ], [ 3, 6, 7, 8, 15, 18 ], [ 3, 6, 7, 8, 16, 17 ], [ 3, 6, 7, 8, 16, 18 ], [ 3, 6, 7, 10, 15, 17 ], [ 3, 6, 7, 10, 15, 18 ], [ 3, 6, 7, 10, 16, 17 ], [ 3, 6, 7, 10, 16, 18 ], [ 3, 7, 8, 15, 16, 17 ], [ 3, 7, 8, 15, 16, 18 ], [ 3, 7, 10, 13, 15, 17 ], [ 3, 7, 10, 13, 15, 18 ], [ 3, 7, 10, 13, 16, 17 ], [ 3, 7, 10, 13, 16, 18 ], [ 3, 9, 10, 13, 15, 17 ], [ 3, 9, 10, 13, 15, 18 ], [ 3, 9, 10, 13, 16, 17 ], [ 3, 9, 10, 13, 16, 18 ], [ 3, 9, 13, 15, 17, 18 ], [ 3, 9, 13, 16, 17, 18 ], [ 3, 12, 14, 15, 16, 17 ], [ 3, 12, 14, 15, 16, 18 ], [ 4, 6, 8, 15, 17, 18 ], [ 4, 6, 8, 16, 17, 18 ], [ 4, 7, 9, 12, 15, 17 ], [ 4, 7, 9, 12, 15, 18 ], [ 4, 7, 9, 12, 16, 17 ], [ 4, 7, 9, 12, 16, 18 ], [ 4, 8, 15, 16, 17, 18 ], [ 4, 9, 10, 12, 15, 17 ], [ 4, 9, 10, 12, 15, 18 ], [ 4, 9, 10, 12, 16, 17 ], [ 4, 9, 10, 12, 16, 18 ], [ 4, 9, 10, 13, 15, 17 ], [ 4, 9, 10, 13, 15, 18 ], [ 4, 9, 10, 13, 16, 17 ], [ 4, 9, 10, 13, 16, 18 ], [ 4, 9, 13, 14, 15, 17 ], [ 4, 9, 13, 14, 15, 18 ], [ 4, 9, 13, 14, 16, 17 ], [ 4, 9, 13, 14, 16, 18 ], [ 4, 10, 12, 15, 17, 18 ], [ 4, 10, 12, 16, 17, 18 ], [ 4, 10, 13, 15, 16, 17 ], [ 4, 10, 13, 15, 16, 18 ], [ 4, 10, 15, 16, 17, 18 ], [ 4, 13, 14, 15, 16, 17 ], [ 4, 13, 14, 15, 16, 18 ], [ 5, 6, 10, 12, 15, 17 ], [ 5, 6, 10, 12, 15, 18 ], [ 5, 6, 10, 12, 16, 17 ], [ 5, 6, 10, 12, 16, 18 ], [ 5, 6, 12, 15, 16, 17 ], [ 5, 6, 12, 15, 16, 18 ], [ 5, 7, 9, 11, 15, 17 ], [ 5, 7, 9, 11, 15, 18 ], [ 5, 7, 9, 11, 16, 17 ], [ 5, 7, 9, 11, 16, 18 ], [ 5, 7, 9, 12, 15, 17 ], [ 5, 7, 9, 12, 15, 18 ], [ 5, 7, 9, 12, 16, 17 ], [ 5, 7, 9, 12, 16, 18 ], [ 5, 7, 11, 15, 17, 18 ], [ 5, 7, 11, 16, 17, 18 ], [ 5, 7, 12, 14, 15, 17 ], [ 5, 7, 12, 14, 15, 18 ], [ 5, 7, 12, 14, 16, 17 ], [ 5, 7, 12, 14, 16, 18 ], [ 5, 7, 14, 15, 17, 18 ], [ 5, 7, 14, 16, 17, 18 ], [ 5, 9, 10, 12, 15, 17 ], [ 5, 9, 10, 12, 15, 18 ], [ 5, 9, 10, 12, 16, 17 ], [ 5, 9, 10, 12, 16, 18 ], [ 5, 12, 14, 15, 16, 17 ], [ 5, 12, 14, 15, 16, 18 ], [ 5, 13, 14, 15, 16, 17 ], [ 5, 13, 14, 15, 16, 18 ], [ 5, 13, 14, 15, 17, 18 ], [ 5, 13, 14, 16, 17, 18 ], [ 6, 7, 8, 15, 17, 18 ], [ 6, 7, 8, 16, 17, 18 ], [ 6, 7, 10, 14, 15, 17 ], [ 6, 7, 10, 14, 15, 18 ], [ 6, 7, 10, 14, 16, 17 ], [ 6, 7, 10, 14, 16, 18 ], [ 6, 7, 14, 15, 17, 18 ], [ 6, 7, 14, 16, 17, 18 ], [ 6, 9, 14, 15, 17, 18 ], [ 6, 9, 14, 16, 17, 18 ], [ 7, 8, 15, 16, 17, 18 ], [ 7, 9, 11, 15, 16, 17 ], [ 7, 9, 11, 15, 16, 18 ], [ 7, 11, 15, 16, 17, 18 ], [ 9, 13, 14, 15, 17, 18 ], [ 9, 13, 14, 16, 17, 18 ], [ 10, 11, 15, 16, 17, 18 ] ]