三维流形上的把柄添加
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摘要
Dehn手术和Heegaard分解是构造三维流形的两种基本方法。这两种方法又都可以通过把柄添加的方式来实现．关于把柄添加方面的一个重要问题是：在双曲流形的亏格至少为2的边界分之上有多少2-把柄添加使所得到的流形是非双曲流形?设M是一个双曲流形，α和β是M的边界分支F上的两条本质的分离简单闭曲线，其中F的亏格至少为2，则已知的结果是：如果M(α)和M(β)是非双曲流形，那么Δ(α，β)≤14。进一步，M.Scharlemann和吴英青(1993)提出了一个猜想：如果M(α)和M(β)是可约的或边界可约的，那么Δ(α，β)=0。本文证明了如果M(α)和M(β)都是可约流形，那么Δ(α，β)≤2。特别地，如果M(α)和M(β)都是可约流形，并且F的亏格为2，那么Δ(α，β)=0。这一结论与参考文献[35]中的结论，以及参考文献[56]中的结论合起来，就得到：在一个双曲三维流形M的一个亏格为2的边界分支F上，最多只有1条分离的简单闭曲线，使得沿其做2-把柄添加能够产生可约流形或边界可约流形。这一结果表明当F的亏格为2时，上面提到的猜想是对的。同时，这一结论还给出了M.Scharlemann关于Refilling问题所提猜想的一个部分证明。
设M_i=V_i∪W_i(i=1，2)是两个不可稳定化的Heegaard分解，且F_i(?)_W_i是(?)_W_i的两个紧致子曲面。同胚映射f：F_1→F_2给出了粘贴流形M=M_1∪_(F_1=F_2)M_2。本文给出了M的Heegaard分解的一种构造方法—曲面连通和，并举了一个例子来说明这种方法构造的Heegaard分解和一般意义下的融合积得到的Heegaard分解是不同的。同时，本文还给出了由曲面连通和得到的Heegaard分解是可稳定化的一个充分条件。
M.Scharlemann和A.Thompson(见[53])给出的thin position的概念在Heegaard分解的研究中起着重要的作用。Heegaard分解的thin position与流形中的双侧可压缩曲面有着密切的关系。本文利用Kneser-Haken定理证明了三维流形中完全不交的双侧可压曲面构成的非过剩集合含有的元素的个数是有上界的；进一步，本文证明了三维流形Heegaard分解的thin position长度的有限性定理。
Dehn surgery and Heegaard splitting are two important methods of constructing 3-manifolds. These two methods can be viewed as handle additions. As an important problemof handle additions, it is interesting to consider that how many 2-handle additions on a genusat least two boundary component of a hyperbolic 3-manifold can obtain a non-hyperbolicmanifold. Let M be a hyperbolic manifold, and F be a component of (?)M of genus at leasttwo. Supposeβandβare two separating slops on F. M. Scharlemann and Y. Wu provedthat if M(α) and M(β) are not hyperbolic, then the intersection numberΔ(α,β)≤14.Furthermore, they gave a conjecture: if M(α) and M(β) are reducible or (?)-reducible, thenΔ(α,β)=0. In this paper, we prove that if both M(α) and M(β) are reducible, thenΔ(α,β)≤2. Specially, if both M(α) and M(β) are reducible and the genus of F is 2, thenΔ(α,β)=0. This result together with the result in [35] and the result in [56] indicates thatthere is only one separating slopαon any component F of (?)M of genus two, such thatM(α) is either reducible or (?)-reducible. This means that the above conjecture is true whenthe genus of F is two. As a corollary of the above results, we also give a partial proof of theRefilling conjecture which is given by M. Scharlemann.
Suppose M_i= V_i∪W_i (i =1, 2) are Heegaard splittings. A homeomorphism f: F_1→F_2produces an attached manifold M= M_1∪_(F_1=F_2)M_2, where F_i(?)_W_i. In this paper wedefine a surface sum of Heegaard splittings induced from the Heegaard splittings of M_1 andM_2. We also give an example showing that the surface sum of Heegaard splittings is differentfrom amalgamation of Heegaard splittings. Furthermore, we give a sufficient condition whenthe surface sum of Heegaard splittings is stabilized.
Thin position of Heegaard splittings defined by M. Scharlemann and A. Thompsonplays an important role in the study of Heegaard splittings. Using Kneser-Haken's Theorem,we prove that any non-excessive set of completely disjoint bicompressible surfaces containsonly finitely elements. Furthermore, we prove that the length of any unstabilized Heegaardsplittings of a 3-manifold has an upper bound.
设M_i=V_i∪W_i(i=1，2)是两个不可稳定化的Heegaard分解，且F_i(?)_W_i是(?)_W_i的两个紧致子曲面。同胚映射f：F_1→F_2给出了粘贴流形M=M_1∪_(F_1=F_2)M_2。本文给出了M的Heegaard分解的一种构造方法—曲面连通和，并举了一个例子来说明这种方法构造的Heegaard分解和一般意义下的融合积得到的Heegaard分解是不同的。同时，本文还给出了由曲面连通和得到的Heegaard分解是可稳定化的一个充分条件。
M.Scharlemann和A.Thompson(见[53])给出的thin position的概念在Heegaard分解的研究中起着重要的作用。Heegaard分解的thin position与流形中的双侧可压缩曲面有着密切的关系。本文利用Kneser-Haken定理证明了三维流形中完全不交的双侧可压曲面构成的非过剩集合含有的元素的个数是有上界的；进一步，本文证明了三维流形Heegaard分解的thin position长度的有限性定理。
Dehn surgery and Heegaard splitting are two important methods of constructing 3-manifolds. These two methods can be viewed as handle additions. As an important problemof handle additions, it is interesting to consider that how many 2-handle additions on a genusat least two boundary component of a hyperbolic 3-manifold can obtain a non-hyperbolicmanifold. Let M be a hyperbolic manifold, and F be a component of (?)M of genus at leasttwo. Supposeβandβare two separating slops on F. M. Scharlemann and Y. Wu provedthat if M(α) and M(β) are not hyperbolic, then the intersection numberΔ(α,β)≤14.Furthermore, they gave a conjecture: if M(α) and M(β) are reducible or (?)-reducible, thenΔ(α,β)=0. In this paper, we prove that if both M(α) and M(β) are reducible, thenΔ(α,β)≤2. Specially, if both M(α) and M(β) are reducible and the genus of F is 2, thenΔ(α,β)=0. This result together with the result in [35] and the result in [56] indicates thatthere is only one separating slopαon any component F of (?)M of genus two, such thatM(α) is either reducible or (?)-reducible. This means that the above conjecture is true whenthe genus of F is two. As a corollary of the above results, we also give a partial proof of theRefilling conjecture which is given by M. Scharlemann.
Suppose M_i= V_i∪W_i (i =1, 2) are Heegaard splittings. A homeomorphism f: F_1→F_2produces an attached manifold M= M_1∪_(F_1=F_2)M_2, where F_i(?)_W_i. In this paper wedefine a surface sum of Heegaard splittings induced from the Heegaard splittings of M_1 andM_2. We also give an example showing that the surface sum of Heegaard splittings is differentfrom amalgamation of Heegaard splittings. Furthermore, we give a sufficient condition whenthe surface sum of Heegaard splittings is stabilized.
Thin position of Heegaard splittings defined by M. Scharlemann and A. Thompsonplays an important role in the study of Heegaard splittings. Using Kneser-Haken's Theorem,we prove that any non-excessive set of completely disjoint bicompressible surfaces containsonly finitely elements. Furthermore, we prove that the length of any unstabilized Heegaardsplittings of a 3-manifold has an upper bound.
引文
[1] J. Berge, The knots in D~2×S~1 with nontrivial Dehn surgery yielding D~2×S~1, Topol. Appl., 38(1991), 1-19.
[2] R. Bing, An Alternative proof that 3-manifolds can be triangulated, Ann. of Math., 69(1959), 37-65.
[3] F. Bonahon and J. Otal, Scindements de Heegaard des espaces lenticularies, Ann. scient. Ec. Norm. Sup., 16(1983), 451-466.
[4] S. Bleiler and M. Scharlemann, A projective plane in R~4 with three critical points is standard. Strongly invertible knots have property P, Topology, 27 (1988), 519 - 540.
[5] S. Bleiler and M. Scharlemann, Tangles, property P, and a problem of J. Martin, Mathematische Annalen, 273(1986), 215-225.
[6] S. Boyer and X. Zhang, Reducing Dehn filling and toroidal Dehn filling, Topol. Appl., 68(1996), 285-303.
[7] G. Burde and H. Zieschang, Knots, Walter de Gruyter, New York, 1985.
[8] A. Casson and C. Gordon, 3-Manifold with irreducible Heegaard splittings of arbitrarily high genus, unpublished.
[9] A. Casson and C. Gordon, Reducing Heegaard splittings, Topology and its applications, 27(1987),275-283.
[10] M. Culler, C. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Ann. Math., 125(1987), 237-300.
[11] M. Eudave Munoz, Primeness and sums of tangle, Trans. Amer. Math. Soc, 306(1988), 773-790.
[12] M. Eudave Munoz, Band sums of links which yield composite links, The cabling conjecture for strongly invertible knots, Trans. Amer. Math. Soc, 330(1992), 463-501.
[13] M. Eudave Munoz, Prime knots obtained by band sums, Pacific J. Math., 139(1989), 53-57.
[14] D. Gabai, On 1-bridge braids in solid tori, Topology, 28(1989), 1-6.
[15] D. Gabai, Genus is superadditive under band connected sum, Topology, 26(1987), 209-210.
[16] C. Gordon, Small surfaces and Dehn fillings, Geom. Topol. Monogr., 2(1999), 177-199.
[17] C. Gordon, Boundary slopes of punctured tori in 3-manifolds, Trans. Amer. Math. Soc, 350(1998), 1713-1790.
[18] C. Gordon and R. Litherland, Incompressible planar surfaces in 3-manifolds, Topol. Appl., 18(1984), 121-144.
[19] C. Gordon and J. Luecke, Reducible manifolds and Dehn surgery, Topology, 35(1996), 385-409.
[20] C. Gordon and J. Luecke, Toroidal and boundary-reducing Dehn fillings, Topology Appl., 93(1999), 77-90.
[21] C. Gordon and J. Luecke, Knots are determined by their complements, Journal of the American Mathematical Society, 2(1989), 371 - 415.
[22] C. Gordon and Y. Wu, Annular and boundary reducing Dehn fillings, Topology, 39(2000), 531-548.
[23] C. Gordon and Y. Wu, Toroidal and annular Dehn fillings, Proc. London. Math. Soc, 78(1999), 662-700.
[24] W. Haken, Some results on surfaces in 3-manifolds, Studies in Modern Topology, MAA. Studies in Math., 5(1968) 39-98.
[25] A. Hatcher On the boundary curves of incompressible surfaces, Pacific J. Math., 99(1982), 373-377.
[26] P. Heegaard, Forstudier til en topologisk teori for de algebraiske fladers sammenhaeng, Dissertation (Cophenhagen, 1898).
[27] J. Hemple, 3-Manifolds, Ann of Math Studies. Princeton Univ. Press Princeton. NJ., 1976.
[28] C. Hayashi and K. Motegi, Only single twists on unknots can produce composite knots, Trans. Amer. Math. Soc, 349(1997), 4465-4479.
[29] C. Hayashi and K. Motegi, Dehn surgery on knots in solid tori creating essential annuli, Trans. Amer. Math. Soc, 349(1997), 4897-4930.
[30] W. Jaco, Lectures on three-manifold topology, Regional Conference Series in Mathematics., 1981.
[31] R. Kirby, Problems in low-dimensional topology, Geometric topology, AMS/IP Stud. Adv. Math., 2.2(Athens, GA, 1993), (Amer. Math. Soc, Providence, RI, 1997), 35-473.
[32] H. Kneser, Geschlossens Flachen in dreidimensionalen Manningfaltigkeiten, Jber. Deutsch. Math.-Verein., 38(1929), 248-260.
[33] T. Kobayashi, Structures of full Haken manifolds, Osaka J.Math., 24(1987), 173-215. [34] M. Lackenby, Attaching handlebodies to 3-manifolds, Geom. Topol., 6(2002), 889 - 904.
[35] Y. Li, R. Qiu and M. Zhang, Boundary reducible handle additions on simple 3-manifolds, arXiv math.GT:0701440.
[36] Tao Li, Heegaard surfaces and measured laminations, I: The Waldhausen conjecture. Invent. Math., 167(2007), 135-177.
[37] Tao Li, Heegaard surfaces and measured laminations, II: non-Haken 3 - manifolds, J. Amer. Math. Soc, 19(2006), 625-657.
[38] K. Morimoto, On the super additivity of tunnel number of knots, Mathematische Annalen, 317(2000), 489-508.
[39] E. Moise, The Triangulation theorem and Hauptvermutung, Ann. of Math., 56(1952), 96-114.
[40] S. Oh, Reducible and toroidal manifolds obtained by Dehn filling, Topol. Appl. 75(1997), 93-104.
[41] R. Qiu, Reducible Dehn surgery and annular Dehn surgery, Pacific J. Math., 192(2000), 357-367.
[42] R. Qiu, Stabilization of Reducible Heegaard Splittings, ArXiv:math.GT/0409497
[43] R. Qiu and J. Ma, Heegaard Splittings of Boundary Reducible 3-Manifolds, ArXiv: math.GT/0409498
[44] R. Qiu and S. Wang, Small knots and large handle additions, Comm. Anal. Geom., 13(2005), 936-961.
[45] R. Qiu and S. Wang, Handle additions producing essential closed surfaces, Pacific J. Math., 229(2007), 233-255.
[46] D. Rolfsen, Knots and Links, Berkeley CA:Publish or Perish Inc, 1976.
[47] M. Scharlemann, Producing reducible 3-manifolds by surgery on a knot, Topology, 29(1990), 481-500.
[48] M. Scharlemann, Refilling meridians in a genus 2 handlebody complement, ArXiv:Mmath.GT/06037005.
[49] M. Scharlemann, Heegaard splittings of compact 3-manifolds, Handbook of geometric topology, ed by R. Daverman and R. Sherr, North-Holland, Amsterdam, 2002, 921-953.
[50] M. Scharlemann, Smooth spheres in R4 with four critical points are standard, Invent. Math., 79(1985)125-141.
[51] M. Scharlemann, Unknotting number one knots are prime, Invent. Math., 82(1985), 37-55.
[52] J. Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc, 35(1933), 88-111.
[53] M. Scharlemann and A. Thompson, Thin position for 3-manifolds, Geometric topology, (Haifa, 1992), 231-238.
[54] M. Scharlemann and A. Thompson, Heegaard Splittings of (Surface) × I Are Standard, Math. Ann., 295(1993), 549-564.
[55] M. Scharlemann and A. Thompson, Unknotting number, genus, and companion tori, Math. Ann., 280(1988), 191-205.
[56] M. Scharlemann and Y. Wu, Hyperbolic manifolds and degenerating handle additions, J. Austral. Math. Soc. Ser. A, 55(1993), 72-89.
[57] F. Waldhausen, Heegaard-Zerlegungen der 3-sphare, Topology, 7(1968), 195-203.
[58] Y. Wu, The reducibility of surgered 3-manifolds, Topology Appl., 43(1992), 213-218.
[59] Y. Wu, Sutured manifold hierarchies, essential laminations, and Dehn surgery, J. Differential Geom., 48(1998), 407-437.
[60] Y. Wu, Dehn fillings producing reducible manifolds and toroidal manifolds, Topology, 37(1998), 95-108.
[61] Y. Wu, Incompressibility of surfaces in surgered 3-manifolds, Topology, 31(1992), 271-279.
[62] M. Zhang, R. Qiu and Y. Li, The distance between two separating, reducing slopes is at most 4, To appear in Math. Z.
[63] 何伯和，廖公夫，基础拓扑学，高等教育出版社，1991．
[64] 尤承业，基础拓扑学讲义，北京大学出版社，2002．
[2] R. Bing, An Alternative proof that 3-manifolds can be triangulated, Ann. of Math., 69(1959), 37-65.
[3] F. Bonahon and J. Otal, Scindements de Heegaard des espaces lenticularies, Ann. scient. Ec. Norm. Sup., 16(1983), 451-466.
[4] S. Bleiler and M. Scharlemann, A projective plane in R~4 with three critical points is standard. Strongly invertible knots have property P, Topology, 27 (1988), 519 - 540.
[5] S. Bleiler and M. Scharlemann, Tangles, property P, and a problem of J. Martin, Mathematische Annalen, 273(1986), 215-225.
[6] S. Boyer and X. Zhang, Reducing Dehn filling and toroidal Dehn filling, Topol. Appl., 68(1996), 285-303.
[7] G. Burde and H. Zieschang, Knots, Walter de Gruyter, New York, 1985.
[8] A. Casson and C. Gordon, 3-Manifold with irreducible Heegaard splittings of arbitrarily high genus, unpublished.
[9] A. Casson and C. Gordon, Reducing Heegaard splittings, Topology and its applications, 27(1987),275-283.
[10] M. Culler, C. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Ann. Math., 125(1987), 237-300.
[11] M. Eudave Munoz, Primeness and sums of tangle, Trans. Amer. Math. Soc, 306(1988), 773-790.
[12] M. Eudave Munoz, Band sums of links which yield composite links, The cabling conjecture for strongly invertible knots, Trans. Amer. Math. Soc, 330(1992), 463-501.
[13] M. Eudave Munoz, Prime knots obtained by band sums, Pacific J. Math., 139(1989), 53-57.
[14] D. Gabai, On 1-bridge braids in solid tori, Topology, 28(1989), 1-6.
[15] D. Gabai, Genus is superadditive under band connected sum, Topology, 26(1987), 209-210.
[16] C. Gordon, Small surfaces and Dehn fillings, Geom. Topol. Monogr., 2(1999), 177-199.
[17] C. Gordon, Boundary slopes of punctured tori in 3-manifolds, Trans. Amer. Math. Soc, 350(1998), 1713-1790.
[18] C. Gordon and R. Litherland, Incompressible planar surfaces in 3-manifolds, Topol. Appl., 18(1984), 121-144.
[19] C. Gordon and J. Luecke, Reducible manifolds and Dehn surgery, Topology, 35(1996), 385-409.
[20] C. Gordon and J. Luecke, Toroidal and boundary-reducing Dehn fillings, Topology Appl., 93(1999), 77-90.
[21] C. Gordon and J. Luecke, Knots are determined by their complements, Journal of the American Mathematical Society, 2(1989), 371 - 415.
[22] C. Gordon and Y. Wu, Annular and boundary reducing Dehn fillings, Topology, 39(2000), 531-548.
[23] C. Gordon and Y. Wu, Toroidal and annular Dehn fillings, Proc. London. Math. Soc, 78(1999), 662-700.
[24] W. Haken, Some results on surfaces in 3-manifolds, Studies in Modern Topology, MAA. Studies in Math., 5(1968) 39-98.
[25] A. Hatcher On the boundary curves of incompressible surfaces, Pacific J. Math., 99(1982), 373-377.
[26] P. Heegaard, Forstudier til en topologisk teori for de algebraiske fladers sammenhaeng, Dissertation (Cophenhagen, 1898).
[27] J. Hemple, 3-Manifolds, Ann of Math Studies. Princeton Univ. Press Princeton. NJ., 1976.
[28] C. Hayashi and K. Motegi, Only single twists on unknots can produce composite knots, Trans. Amer. Math. Soc, 349(1997), 4465-4479.
[29] C. Hayashi and K. Motegi, Dehn surgery on knots in solid tori creating essential annuli, Trans. Amer. Math. Soc, 349(1997), 4897-4930.
[30] W. Jaco, Lectures on three-manifold topology, Regional Conference Series in Mathematics., 1981.
[31] R. Kirby, Problems in low-dimensional topology, Geometric topology, AMS/IP Stud. Adv. Math., 2.2(Athens, GA, 1993), (Amer. Math. Soc, Providence, RI, 1997), 35-473.
[32] H. Kneser, Geschlossens Flachen in dreidimensionalen Manningfaltigkeiten, Jber. Deutsch. Math.-Verein., 38(1929), 248-260.
[33] T. Kobayashi, Structures of full Haken manifolds, Osaka J.Math., 24(1987), 173-215. [34] M. Lackenby, Attaching handlebodies to 3-manifolds, Geom. Topol., 6(2002), 889 - 904.
[35] Y. Li, R. Qiu and M. Zhang, Boundary reducible handle additions on simple 3-manifolds, arXiv math.GT:0701440.
[36] Tao Li, Heegaard surfaces and measured laminations, I: The Waldhausen conjecture. Invent. Math., 167(2007), 135-177.
[37] Tao Li, Heegaard surfaces and measured laminations, II: non-Haken 3 - manifolds, J. Amer. Math. Soc, 19(2006), 625-657.
[38] K. Morimoto, On the super additivity of tunnel number of knots, Mathematische Annalen, 317(2000), 489-508.
[39] E. Moise, The Triangulation theorem and Hauptvermutung, Ann. of Math., 56(1952), 96-114.
[40] S. Oh, Reducible and toroidal manifolds obtained by Dehn filling, Topol. Appl. 75(1997), 93-104.
[41] R. Qiu, Reducible Dehn surgery and annular Dehn surgery, Pacific J. Math., 192(2000), 357-367.
[42] R. Qiu, Stabilization of Reducible Heegaard Splittings, ArXiv:math.GT/0409497
[43] R. Qiu and J. Ma, Heegaard Splittings of Boundary Reducible 3-Manifolds, ArXiv: math.GT/0409498
[44] R. Qiu and S. Wang, Small knots and large handle additions, Comm. Anal. Geom., 13(2005), 936-961.
[45] R. Qiu and S. Wang, Handle additions producing essential closed surfaces, Pacific J. Math., 229(2007), 233-255.
[46] D. Rolfsen, Knots and Links, Berkeley CA:Publish or Perish Inc, 1976.
[47] M. Scharlemann, Producing reducible 3-manifolds by surgery on a knot, Topology, 29(1990), 481-500.
[48] M. Scharlemann, Refilling meridians in a genus 2 handlebody complement, ArXiv:Mmath.GT/06037005.
[49] M. Scharlemann, Heegaard splittings of compact 3-manifolds, Handbook of geometric topology, ed by R. Daverman and R. Sherr, North-Holland, Amsterdam, 2002, 921-953.
[50] M. Scharlemann, Smooth spheres in R4 with four critical points are standard, Invent. Math., 79(1985)125-141.
[51] M. Scharlemann, Unknotting number one knots are prime, Invent. Math., 82(1985), 37-55.
[52] J. Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc, 35(1933), 88-111.
[53] M. Scharlemann and A. Thompson, Thin position for 3-manifolds, Geometric topology, (Haifa, 1992), 231-238.
[54] M. Scharlemann and A. Thompson, Heegaard Splittings of (Surface) × I Are Standard, Math. Ann., 295(1993), 549-564.
[55] M. Scharlemann and A. Thompson, Unknotting number, genus, and companion tori, Math. Ann., 280(1988), 191-205.
[56] M. Scharlemann and Y. Wu, Hyperbolic manifolds and degenerating handle additions, J. Austral. Math. Soc. Ser. A, 55(1993), 72-89.
[57] F. Waldhausen, Heegaard-Zerlegungen der 3-sphare, Topology, 7(1968), 195-203.
[58] Y. Wu, The reducibility of surgered 3-manifolds, Topology Appl., 43(1992), 213-218.
[59] Y. Wu, Sutured manifold hierarchies, essential laminations, and Dehn surgery, J. Differential Geom., 48(1998), 407-437.
[60] Y. Wu, Dehn fillings producing reducible manifolds and toroidal manifolds, Topology, 37(1998), 95-108.
[61] Y. Wu, Incompressibility of surfaces in surgered 3-manifolds, Topology, 31(1992), 271-279.
[62] M. Zhang, R. Qiu and Y. Li, The distance between two separating, reducing slopes is at most 4, To appear in Math. Z.
[63] 何伯和，廖公夫，基础拓扑学，高等教育出版社，1991．
[64] 尤承业，基础拓扑学讲义，北京大学出版社，2002．