The Maps From Some Special 3-manifolds To S~2 And Their Triangulations

There are many methods to construct 3-dimensional manifolds.In this paper,some maps from 3-dimensional manifolds to 2-dimensional spheres and their triangulations are given by special methods.The 3-dimensional real projective space RP3 can be obtained by gluing the radial points of the 3-dimensional sphere S3.Let r:RP3?S2 be a map from RP3 to S2 and be induced by Hopf map.RP3 can also be decompose into the union of two subspaces with common boundary by Heegaard splitting,denoted as RP3=V3?T2 W3 The triangulation RP123 of RP3 is obtained by means of the triangulation of the two subspaces.So,we can define a simplicial map r';RP123?S42,and prove the simplicial map r' is a triangulation of the map r.By Heegaard splitting,let Vg,Wg be handlebodies of genus g,and let f be an orientation reversing homeomorphism from the boundary of Vg to the boundary of Wg.By gluing Vg,Wg along f we obtain the compact oriented 3-manifold M=Vg?f Wg.Every closed,orientable 3-manifold may be so obtained.Let h:Sg?S1 be a map from Sg to S1.It can be extended to Vg,Wg,then we can get a map H:M?S2.We first consider triangulations of the two handlebodies,and then we can obtain a triangulation of the 3-dimensional manifold M that is M12g.We consider the triangulation of map h:Sg?S1 that is h':(Sg)9+10(g-1)?S31,then we can obtain a simplicial map H':M12g?S42 that is a triangulation of the map H:M?S2.3-dimensional sphere is a union of the infinity point {?} join with the 2-dimensional sphere S2=(?)(D3)and the solid 3-dimensional sphere D3,denoted as S3=((?)(D3)*{?})?id D3.The map id:S2?S2 is a identity map.So we can get triangulations of S3 from triangulations of S2,D3.Let S?3 be a triangulation of S3.For(?)d ? N*,there are simplicial self-maps f:S?3(d)?S53 of degree d.The triangulations of D3 and S2 are used to construct the triangulation of S3 and the simplicial self-maps.And we can get the upper bound of ?(d).