The Relative Equivalence Of Separating Curves On Surfaces And Its Applications To Haken Spheres

In 1950's, Moise and Bing proved that each manifold whose dimention≤3 has the piecewise linear structure, and from their results that any closed connected and orientable 3-manifold M is homeomorphism to the polyhedron \K\ of a simplicial complex K, we know that M has Heegaard splittings.Let V Uf W be a Heegaard splitting of M. Let 5 be a separating 2-sphere in M, if S intersects the Heegaard surface F only in an essential circle in F, we say S is a Haken sphere (with respect to V Uf W), C = S n F is a Haken cruve.Haken spheres are important in 3-manifold. in [40], Scharlemann and Thonpson describe how two Haken spheres in a genus 2 Heegaard splitting of S~3are related, in [24], Lei describe how two Haken spheres in a genus 2 Heeaard splitting of a connected sum of two lens spaces, in [26], Lei and zhang describe how two Haken spheres in a genus 2 Heegaard splitting of S~2 ×S~1#S~2 × S~1 or of S~2×S~1#L(p, q).This paper, we first work in the research of the separating simple closed cruvesof the connected orientable closed surface with respect to one patition of a complete curves system of the surface , then we apply the results to the Haken spheres. We work in the piecewise linear categary and we consider the connected orientable closed 3-manifold.We denote F be the connected orientable closed surface with genus n(n > 2).Let X = {Ai,...,Ap,Bi,...,Bq} where p + q = n,p,q > 0, be a complete system of F. Lets*' = {A\,..., Ap}, SB = {Bi,..., Bq}, we say (^, 38) be a partition of E.Let C be an essential separating simple closed curve on F. if C cuts F into two pieces, sayFi,F%, such that si C F\,SB C F2, we say C is separating with respect toLet C\ and C% be two disjoint simple closed curve on F, a be a simple arc on F, such that a connects C\ and C2 and the interior of a is disjoint from Ciand C2, Let N = N{C\ U C2 U q) be a small regular neighborhood of C\ U C2 U a on F. Then the component of dN which is other than the copies of C\ and C-i is called the band sum of C\ and C2 and is denoted by Cf2 = Ci#aC2-Similarly, Let A = {C\,... ,Cm} be a set of pairwise disjoint simple closed curves, and A = {ax,..., am_i be a set of pairwise disjoint simple arcs on F, A and A satisfy the above condition, then we have the band sum C^ = Ci#aiC2#a2... #am-iLet C and 7 be two simple closed curves on F. If C intersects 7 essentially into two points and C is disjoint from the partition (stf, 88) of the complete curve systemE, we say that 7 be the adjoint curve of C with respect to the partition (srf, &).Let iV(7) be a neighborhood of 7 on F, then iV(7) = 7 x [-1,1]. Let h-, : N(y) -> N(*y) such that /^(fl, y) = (0 + n(y + 1), y). It is clearly that /^ keeps the points of 9jV(7) fixed, and twists it in the interior of ^(7). We can extend h^ to the homeomorphism /i7 : F —* F, such that /i-yl^Cy) = h~r and ^I'J fcybea Dehn twist of C along 7 on F.If we consider F be the Heegaard surface of the Heegaard splitting M = let S be a Haken sphere in V Uj? W and let C be the corresponding Haken curve. If we denote D = V n 5,￡ = Wn S,then V = Vi UD V^W = Wa U￡ W2. let ?f, F by making use of the Dhen twist along 7 , which is the adjoint curve of C with respect to the papition (A, B),then A"(C),Vn 6 Z is separating with respect to the patition (A, B) of E ,and each separating curve with respect to the partition (A, B) of E , say C , must isotopic to h^(C),n￡Z .