三维流形融合积的Heegaard分解
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摘要
流形的分类是研究流形的重要课题之一,由于Kneser-Milnor定理,JSJ分解定理以及Thurston几何化猜想的提出,使得人们更加关注三维流形,并相信三维流形是可以完全拓扑分类的.
从另一个角度研究三维流形,就是将两个三维流形沿着同胚的边界相粘,所得流形我们称之为三维流形的融合积.一般地,我们只考虑Haken流形的融合积.
如果一个闭曲面S将三维流形M分成两个压缩体V和W,那么我们称V∪S W是M的一个Heegaard分解.由于任意一个紧致流形都存在Heegaard分解,Heegaard分解日渐成为组合三维流形的一个重要不变量,对三维流形的分类起着重要作用.但是,并不像我们期望的那样,任一三维流形的Heegaard分解都是唯一的.三维流形Heegaard分解的唯一性只是一种极为特殊的现象.
本文主要研究了三维流形融合积的Heegaard分解,给出了三维流形自融合积的Heegaard分解如果满足一定条件,他的极小Heegaard分解在合痕意义下具有唯一性,并给出了三维流形融合积的Heegaard分解非退化的一个最好的条件.
三维流形融合积的Heegaard分解是将两个流形沿闭曲面相粘,本文还讨论了将两个三维流形沿平环相粘,给出了他的具体的Heegaard结构,并证明了如果有一个平环是非分离的,那么我们有g(M)=g(M1)+g(M2).此结果蕴含了纽结的一个组合不变量tunnel number的超可加性.
The classification of manifolds is an important topic of studying manifolds. Because of Kneser-Milnor theorem, JSJ decomposition theorem and the Thurston geometrization conjecture, now people are interested in 3-manifold, and believed that 3-manifolds can be completely topological classification.
From another point of view, we call 3-manifold obtained by gluing two 3-manifolds along their homeomorphic boundary the amalgamated 3-manifold. In general, we consider only the amalgamated Haken 3-manifold.
If a closed surface S cuts a 3-manifold into two compression bodies V and W, then we call V Us W the Heegaard splitting of M.
Since any 3-manifold has a Heegaard splitting, it becomes an important invariant of combinatorial 3-manifolds and play an important role for classifying 3-manifolds. However, it is not as we expected, any Heegaard splitting of 3-manifold is unique. The uniqueness of Heegaard splitting of 3-manifold is only a very special phenomenon.
We mainly discuss the Heegaard splittings of amalgamated 3-manifold, prove that if the self-amalgamated 3-manifold satisfies some conditions, then the minimal Heegaard splitting is unique up to isotopy. We still show the best condition of amalgamated 3-manifold which the Heegaard genus is not degenerate.
The amalgamated 3-manifold is obtained by gluing two closed surfaces. In this paper, we also discuss the 3-manifolds obtained by gluing annuli, prove that if one of annuli is not separating, then we have g(M)= g(M1)+g(M2). This result implies a combinatorial knot invariant:the super additivity of tunnel number.
从另一个角度研究三维流形,就是将两个三维流形沿着同胚的边界相粘,所得流形我们称之为三维流形的融合积.一般地,我们只考虑Haken流形的融合积.
如果一个闭曲面S将三维流形M分成两个压缩体V和W,那么我们称V∪S W是M的一个Heegaard分解.由于任意一个紧致流形都存在Heegaard分解,Heegaard分解日渐成为组合三维流形的一个重要不变量,对三维流形的分类起着重要作用.但是,并不像我们期望的那样,任一三维流形的Heegaard分解都是唯一的.三维流形Heegaard分解的唯一性只是一种极为特殊的现象.
本文主要研究了三维流形融合积的Heegaard分解,给出了三维流形自融合积的Heegaard分解如果满足一定条件,他的极小Heegaard分解在合痕意义下具有唯一性,并给出了三维流形融合积的Heegaard分解非退化的一个最好的条件.
三维流形融合积的Heegaard分解是将两个流形沿闭曲面相粘,本文还讨论了将两个三维流形沿平环相粘,给出了他的具体的Heegaard结构,并证明了如果有一个平环是非分离的,那么我们有g(M)=g(M1)+g(M2).此结果蕴含了纽结的一个组合不变量tunnel number的超可加性.
The classification of manifolds is an important topic of studying manifolds. Because of Kneser-Milnor theorem, JSJ decomposition theorem and the Thurston geometrization conjecture, now people are interested in 3-manifold, and believed that 3-manifolds can be completely topological classification.
From another point of view, we call 3-manifold obtained by gluing two 3-manifolds along their homeomorphic boundary the amalgamated 3-manifold. In general, we consider only the amalgamated Haken 3-manifold.
If a closed surface S cuts a 3-manifold into two compression bodies V and W, then we call V Us W the Heegaard splitting of M.
Since any 3-manifold has a Heegaard splitting, it becomes an important invariant of combinatorial 3-manifolds and play an important role for classifying 3-manifolds. However, it is not as we expected, any Heegaard splitting of 3-manifold is unique. The uniqueness of Heegaard splitting of 3-manifold is only a very special phenomenon.
We mainly discuss the Heegaard splittings of amalgamated 3-manifold, prove that if the self-amalgamated 3-manifold satisfies some conditions, then the minimal Heegaard splitting is unique up to isotopy. We still show the best condition of amalgamated 3-manifold which the Heegaard genus is not degenerate.
The amalgamated 3-manifold is obtained by gluing two closed surfaces. In this paper, we also discuss the 3-manifolds obtained by gluing annuli, prove that if one of annuli is not separating, then we have g(M)= g(M1)+g(M2). This result implies a combinatorial knot invariant:the super additivity of tunnel number.
引文
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[3]Milnor J. A unique factorization theorem for 3-manifolds [J]. Amer. J. Math.,1962, 84:1-7.
[4]Jaco W., Shalen P. Seifert fibered spaces in 3-manifolds [C]//Geometric topology (Proc. Georgia Topology Conf., Athens, Ga.,1977). New York-London:Academic Press,1979:91-99.
[5]Johannson K. On exotic homotopy equivalences of 3-manifolds [C]//Geometric topol-ogy (Proc. Georgia Topology Conf., Athens, Ga.,1977). New York-London:Academic Press,1979:101-111.
[6]Thurston W. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry [J]. Bull. Amer. Math. Soc. (N.S.),1982,6(3):357-381.
[7]Thurston W. Three-dimensional geometry and topology Vol.1. Edited by Silvio Levy. Princeton Mathematical Series,35. [M]. Princeton:Princeton University Press,1997.
[8]Thurston W. The Geometry and Topology of Three-Manifolds [M]. Princeton lecture notes on geometric structures on 3-manifolds,1980.
[9]Perelman G. The entropy formula for the Ricci flow and its geometric applications [J]. ArXiv:math.DG/0211159.
[10]Perelman G. Ricci flow with surgery on three-manifolds [J]. ArXiv:math.DG/0303109.
[11]Perelman G. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds [J]. ArXiv:math.DG/0307245.
[12]Morimoto K. On the super additivity of tunnel number of knots [J]. Math. Ann.,2000, 317(3):489-508.
[13]Morimoto K. Characterization of tunnel number two knots which have the property "2+1= 2" [J]. Topology Appl.,1995,64(2):165-176.
[14]Morimoto K., Sakuma M., Yokota Y. Examples of tunnel number one knots which have the property "1+1= 3" [J] Math. Proc. Cambridge Philos. Soc.,1996,119(1):113-118.
[15]Kobayashi T. A construction of arbitrarily high degeneration of tunnel numbers of knots under connected sum [J] J. Knot Theory Ramifications,1994,3(2):179-186.
[16]Moriah Y., Rubinstein H. Heegaard structures of negatively curved 3-manifolds [J]. Comm. Anal. Geom.1997,5(3):375-412.
[17]Morimoto K. On the additivity of tunnel number of knots [J]. Topology Appl.,1993, 53(1):37-66.
[18]Kobayashi T., Rieck Y. Knot exteriors with additive Heegaard genus and Morimoto's conjecture [J]. Algebr. Geom. Topol.,2008,8(2):953-969.
[19]Kobayashi T., Rieck Y. Knots with g(E(K))= 2 and g(E(K#K#K))= 6 and Mori-moto's conjecture [J]. Topology Appl.,2009,156(6):1114-1117.
[20]Qiu R., Du K., Ma J., Zhang M. Heegaard genera of annular 3-manifolds [J]. To appear J. Knot Theory Ramifications.
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[25]Qiu R. Untelescopings of Heegaard splittings [J]. Northeast. Math. J.,2000,16(4):484-490.
[26]Casson A., Gordon C. Reducing Heegaard splittings [J]. Topology and its applications, 1987,27:275-283.
[27]Scharlemann M., Thompson A. Thin position for 3-manifolds [M]. Geometric topology, Haifa,1992,231-238.
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[31]Lei F. On stability of Heegaard splittings [J]. Math. Proc. Cambridge Philos. Soc. 2000,129(1):55-57.
[32]F. Waldhausen, Heegaard-Zerlegungen der 3-sphare [J]. Topology,1968,7:195-203.
[33]F. Bonahon, Diffeotopies des espaces lenticulaires [J]. Topology,1983,22:305-314.
[34]Bonahon F., Otal J. Scindement de Heegaard des espaces lenticulaires [J]. C. R. Acad. Sci. Paris Ser. I Math.,1982,294:585-587.
[35]Scharlemann M., Thompson A. Heegaard Splittings of (Surface)×I Are Standard [J]. Math. Ann.,1993,295:549-564.
[36]Schultens J. The classification of Heegaard splittings for (compact orientable surface) × S1 [J]. Proc. London Math. Soc.,1993,67(3(2)):425-448.
[37]Frohman C., Hass J. Unstable minimal surfaces and Heegaard splittings [J]. Invent. Math.1989,95:529-540.
[38]Boileau M., Otal J. Sur les scindements de Heegaard du tore T3 [J]. J. Differential Geom.,1990,32(1):209-233.
[39]Birman J., Gonzalez-Acuna F., Montesinos J. Heegaard splittings of prime manifolds are not unique [J]. Mich. Math. J.,1976,23:97-103.
[40]Casson A., Gordon C.3-Manifold with irreducible Heegaard splittings of arbitrarily high genus [J]. Unpublished.
[41]Kobayashi T. There exist 3-manifolds with arbitrarily high genus irreducible Heegaard splittings [J]. Unpublished.
[42]Waldhausen F. Some problems on 3-manifolds [C]. Proc. Sympos. Pure Math., Amer. Math. Soc., Providence,1978,32:313-322.
[43]Li T. Heegaard surfaces and measured laminations, I:The Waldhausen conjecture [J]. Invent. Math.,2007,167:135-177.
[44]Li T. Heegaard surfaces and measured laminations, II:non-Haken 3-manifolds [J]. J. Amer.Math. Soc.,2006,19:625-657.
[45]Kobayashi T., Qiu R., Rieck Y., Wang S., Separating incompressible surfaces and stabilizations of Heegaard splittings [J]. Math. Proc. Cambridge Philos. Soc.,2004, 137:633-643.
[46]Schultens J., Weidmann R. Destabilizing amalgamated Heegaard splittings. [M]. Work-shop on Heegaard Splittings,319-334, Geom. Topol. Monogr., Geom. Topol. Publ., 2007,12:319-334.
[47]Lackenby M. The Heegaard genus of amalgamated 3-manifolds [J]. Geom. Dedicata, 2004,109:139-145.
[48]Bachman D., Schleimer S., Sedgwick E. Sweepouts of amalgamated 3-manifolds [J]. Algebr. Geom. Topol.,2006,6:171-194.
[49]Li T. On the Heegaard splittings of amalgamated 3-manifolds [J]. Geom. Topol. Mono-graphs,2007,12:157-190.
[50]Souto J. The Heegaard genus and distance in curve complex [J]. Preprint.
[51]Kobayashi T., Qiu R. The amalgamation of high distance Heegaard splittings is always efficient [J]. Math. Ann.,2008,341(3):707-715.
[52]Lei F., Yang G. A lower bound of genus of amalgamations of Heegaard splittings [J]. Math. Proc. Cambridge Philos. Soc.,2009,146(3):615-623.
[53]Yang G., Lei F. On amalgamations of Heegaard splittings with high distance [J]. Proc. Amer. Math. Soc.,2009,137(2):723-731.
[54]Yang G., Lei F. Amalgamations of Heegaard splittings in 3-manifolds without some essential surfaces [J]. To appear Albegraic & Geometric Topology.
[55]Du K., Qiu R. The self-amalgamation of high distance Heegaard splittings is always efficient [J]. To appear Topology and its applications.
[56]Du K., Gao X. A note on Heegaard splittings of amalgamated 3-manifolds [J]. Preprint.
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[2]Kneser H. Geschlossene Flachen in dreidimensionalen Mannigfaltigkeiten [J]. Jahres-bericht der Deut. Math. Verein.,1929,38:248-260.
[3]Milnor J. A unique factorization theorem for 3-manifolds [J]. Amer. J. Math.,1962, 84:1-7.
[4]Jaco W., Shalen P. Seifert fibered spaces in 3-manifolds [C]//Geometric topology (Proc. Georgia Topology Conf., Athens, Ga.,1977). New York-London:Academic Press,1979:91-99.
[5]Johannson K. On exotic homotopy equivalences of 3-manifolds [C]//Geometric topol-ogy (Proc. Georgia Topology Conf., Athens, Ga.,1977). New York-London:Academic Press,1979:101-111.
[6]Thurston W. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry [J]. Bull. Amer. Math. Soc. (N.S.),1982,6(3):357-381.
[7]Thurston W. Three-dimensional geometry and topology Vol.1. Edited by Silvio Levy. Princeton Mathematical Series,35. [M]. Princeton:Princeton University Press,1997.
[8]Thurston W. The Geometry and Topology of Three-Manifolds [M]. Princeton lecture notes on geometric structures on 3-manifolds,1980.
[9]Perelman G. The entropy formula for the Ricci flow and its geometric applications [J]. ArXiv:math.DG/0211159.
[10]Perelman G. Ricci flow with surgery on three-manifolds [J]. ArXiv:math.DG/0303109.
[11]Perelman G. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds [J]. ArXiv:math.DG/0307245.
[12]Morimoto K. On the super additivity of tunnel number of knots [J]. Math. Ann.,2000, 317(3):489-508.
[13]Morimoto K. Characterization of tunnel number two knots which have the property "2+1= 2" [J]. Topology Appl.,1995,64(2):165-176.
[14]Morimoto K., Sakuma M., Yokota Y. Examples of tunnel number one knots which have the property "1+1= 3" [J] Math. Proc. Cambridge Philos. Soc.,1996,119(1):113-118.
[15]Kobayashi T. A construction of arbitrarily high degeneration of tunnel numbers of knots under connected sum [J] J. Knot Theory Ramifications,1994,3(2):179-186.
[16]Moriah Y., Rubinstein H. Heegaard structures of negatively curved 3-manifolds [J]. Comm. Anal. Geom.1997,5(3):375-412.
[17]Morimoto K. On the additivity of tunnel number of knots [J]. Topology Appl.,1993, 53(1):37-66.
[18]Kobayashi T., Rieck Y. Knot exteriors with additive Heegaard genus and Morimoto's conjecture [J]. Algebr. Geom. Topol.,2008,8(2):953-969.
[19]Kobayashi T., Rieck Y. Knots with g(E(K))= 2 and g(E(K#K#K))= 6 and Mori-moto's conjecture [J]. Topology Appl.,2009,156(6):1114-1117.
[20]Qiu R., Du K., Ma J., Zhang M. Heegaard genera of annular 3-manifolds [J]. To appear J. Knot Theory Ramifications.
[21]Heegaard P. Forstudier til en topologisk teori for de algebraiske fladers sammenhaeng [J]. Dissertation, Cophenhagen,1898.
[22]Bing R. An Alternative proof that 3-manifolds can be triangulated [J]. Ann. of Math., 1959,69:37-65.
[23]Moise E. The Triangulation theorem and Hauptvermutung [J]. Ann. of Math.,1952, 56:96-114.
[24]Haken W. Some results on surfaces in 3-manifolds [M]. Studies in Modern Topology, MAA. Studies in Math.,1968,5:39-98.
[25]Qiu R. Untelescopings of Heegaard splittings [J]. Northeast. Math. J.,2000,16(4):484-490.
[26]Casson A., Gordon C. Reducing Heegaard splittings [J]. Topology and its applications, 1987,27:275-283.
[27]Scharlemann M., Thompson A. Thin position for 3-manifolds [M]. Geometric topology, Haifa,1992,231-238.
[28]Milnor J. Morse theory [C]//Annals of Mathematics Studies(Based on lecture notes by M. Spivak and R. Wells). Princeton:Princeton University 1979(51).
[29]Reidemeister K. Zur dreidimensionalen Topologie [J]. Abh. Math. Sem. Univ. Hamburg, 1936,9:189-194.
[30]Singer J. Three-dimensional manifolds and their Heegaard-diagram [J]. Trans. Amer. Math. Soc.,1933,35:88-111.
[31]Lei F. On stability of Heegaard splittings [J]. Math. Proc. Cambridge Philos. Soc. 2000,129(1):55-57.
[32]F. Waldhausen, Heegaard-Zerlegungen der 3-sphare [J]. Topology,1968,7:195-203.
[33]F. Bonahon, Diffeotopies des espaces lenticulaires [J]. Topology,1983,22:305-314.
[34]Bonahon F., Otal J. Scindement de Heegaard des espaces lenticulaires [J]. C. R. Acad. Sci. Paris Ser. I Math.,1982,294:585-587.
[35]Scharlemann M., Thompson A. Heegaard Splittings of (Surface)×I Are Standard [J]. Math. Ann.,1993,295:549-564.
[36]Schultens J. The classification of Heegaard splittings for (compact orientable surface) × S1 [J]. Proc. London Math. Soc.,1993,67(3(2)):425-448.
[37]Frohman C., Hass J. Unstable minimal surfaces and Heegaard splittings [J]. Invent. Math.1989,95:529-540.
[38]Boileau M., Otal J. Sur les scindements de Heegaard du tore T3 [J]. J. Differential Geom.,1990,32(1):209-233.
[39]Birman J., Gonzalez-Acuna F., Montesinos J. Heegaard splittings of prime manifolds are not unique [J]. Mich. Math. J.,1976,23:97-103.
[40]Casson A., Gordon C.3-Manifold with irreducible Heegaard splittings of arbitrarily high genus [J]. Unpublished.
[41]Kobayashi T. There exist 3-manifolds with arbitrarily high genus irreducible Heegaard splittings [J]. Unpublished.
[42]Waldhausen F. Some problems on 3-manifolds [C]. Proc. Sympos. Pure Math., Amer. Math. Soc., Providence,1978,32:313-322.
[43]Li T. Heegaard surfaces and measured laminations, I:The Waldhausen conjecture [J]. Invent. Math.,2007,167:135-177.
[44]Li T. Heegaard surfaces and measured laminations, II:non-Haken 3-manifolds [J]. J. Amer.Math. Soc.,2006,19:625-657.
[45]Kobayashi T., Qiu R., Rieck Y., Wang S., Separating incompressible surfaces and stabilizations of Heegaard splittings [J]. Math. Proc. Cambridge Philos. Soc.,2004, 137:633-643.
[46]Schultens J., Weidmann R. Destabilizing amalgamated Heegaard splittings. [M]. Work-shop on Heegaard Splittings,319-334, Geom. Topol. Monogr., Geom. Topol. Publ., 2007,12:319-334.
[47]Lackenby M. The Heegaard genus of amalgamated 3-manifolds [J]. Geom. Dedicata, 2004,109:139-145.
[48]Bachman D., Schleimer S., Sedgwick E. Sweepouts of amalgamated 3-manifolds [J]. Algebr. Geom. Topol.,2006,6:171-194.
[49]Li T. On the Heegaard splittings of amalgamated 3-manifolds [J]. Geom. Topol. Mono-graphs,2007,12:157-190.
[50]Souto J. The Heegaard genus and distance in curve complex [J]. Preprint.
[51]Kobayashi T., Qiu R. The amalgamation of high distance Heegaard splittings is always efficient [J]. Math. Ann.,2008,341(3):707-715.
[52]Lei F., Yang G. A lower bound of genus of amalgamations of Heegaard splittings [J]. Math. Proc. Cambridge Philos. Soc.,2009,146(3):615-623.
[53]Yang G., Lei F. On amalgamations of Heegaard splittings with high distance [J]. Proc. Amer. Math. Soc.,2009,137(2):723-731.
[54]Yang G., Lei F. Amalgamations of Heegaard splittings in 3-manifolds without some essential surfaces [J]. To appear Albegraic & Geometric Topology.
[55]Du K., Qiu R. The self-amalgamation of high distance Heegaard splittings is always efficient [J]. To appear Topology and its applications.
[56]Du K., Gao X. A note on Heegaard splittings of amalgamated 3-manifolds [J]. Preprint.
[57]Hatcher A. Algebraic topology [M]. Cambridge:Cambridge University Press,2002.
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