INFINITE-DIMENSIONAL ALGEBRAS IN DIMENSIONALLY REDUCED STRING THEORY

a rXiv:h ep-th/98781v11J ul1998UICHEP-TH/98-7,February 1,2008Algebraic Shape Invariant Models S.Chaturvedi a,1,R.Dutt b,2,A.Gangopadhyaya c,3,P.Panigrahi a,4,C.Rasinariu d,5and U.Sukhatme d,6a)Department of Physics,University of Hyderabad,Hyderabad,India;b)Department of Physics,Visva Bharati University,Santiniketan,India;c)Department of Physics,Loyola University Chicago,Chicago,USA;d)Department of Physics,University of Illinois at Chicago,Chicago,USA.Abstract Motivated by the shape invariance condition in supersymmetric quantum mechanics,we develop an algebraic framework for shape invariant Hamiltonians with a general change of pa-rameters.This approach involves nonlinear generalizations of Lie algebras.Our work extends previous results showing the equivalence of shape invariant potentials involving translational change of parameters with standard SO (2,1)potential algebra for Natanzon type potentials.In recent years,there has been considerable interest in studying exactly solvable quantum mechanical problems using algebraic approaches[1-9].In this respect,supersymmetric quantum mechanics(SUSYQM)has been found to be an elegant and useful prescription for obtaining closed analytic expressions both for the energy eigenvalues and eigenfunctions for a large class of one dimensional(or spherically symmetric three dimensional)problems[3].The main ingredients in SUSYQM are the supersymmetric partner Hamiltonians H−≡A†A and H+≡AA†.The A and A†operators used in this factorization are expressed in terms of the real superpotential W as follows:A(x,a)=ddx+W(x,a).(1)Here,a is a parameter(or a set of parameters),which will play an important role in the approach of this paper.An interesting feature of SUSYQM is that for a shape invariant system[1-3],i.e.a system satisfyingH+(x,a0)=H−(x,a1)+R(a0);a1=f(a0),(2) the entire spectrum can be determined algebraically without ever referring to underlying differ-ential equations.Recently,it has been shown that shape invariant potentials in which the change of parameters a1=f(a0)is of translational type a1=a0+k,k=constant,possess a SO(2,1)potential algebra and are solvable by group theoretical techniques[4-7].Thus,for such potentials,there appear to be two seemingly independent algebraic methods,SUSYQM and potential algebras,for obtaining the complete spectra.One may naturally ask the question whether there is any connection between the two methods.Indeed,the equivalence between the two algebraic approaches has been established[6].More specifically,for all shape invariant potentials of the Natanzon type [10],the underlying potential algebra is SO(2,1).Other similar approaches are discussed in refs. [4,11].The purpose of this letter is to examine whether such an equivalence is also valid for a larger class of shape invariant potentials.We consider a much more general class of change of param-eters,which contains both translations as well as scalings.In particular,for shape invariant potentials generated by a pure scaling ansatz[12]a1=qa0,q=constant(0<q<1),wefind that the associated potential algebra is a nonlinear deformation of SO(2,1).In fact,coordinate representations of these nonlinear algebras are an interesting area of research in their own right from a mathematical point of view[13].We also show that the energy spectrum resulting from2this potential algebra agrees with the results of SUSYQM.To begin the construction of the operator algebra,let us express eq.(2)in terms of A and A†:A(x,a0)A†(x,a0)−A†(x,a1)A(x,a1)=R(a0).(3) This relation resembles a commutator structure.To obtain a closed SO(2,1)type Lie algebra, we introduce an auxiliary variableφand defineJ+=Q(φ)A†(x,χ(i∂φ)),J−=A(x,χ(i∂φ))Q⋆(φ),(4)where Q(φ)is a function to be determined andχis an arbitrary,real function.The operator A(x,χ(i∂φ))is given by eq.(1)with the substitutiona→χ(i∂φ).(5) From eq.(4),one obtains[J+,J−]=Q(φ)A†(x,χ(i∂φ))A(x,χ(i∂φ))Q⋆(φ)−A(x,χ(i∂φ))Q⋆(φ)Q(φ)A†(x,χ(i∂φ)).(6)Choosing Q(φ)=e ipφ,where p is an arbitrary real constant,eq.(6)can be easily cast into the following form[J+,J−]=− A(x,χ(i∂φ))A†(x,χ(i∂φ))−A†(x,χ(i∂φ+p))A(x,χ(i∂φ+p)) .(7) In analogy with eq.(3),we impose the algebraic shape invariance conditionA(x,χ(i∂φ))A†(x,χ(i∂φ))−A†(x,χ(i∂φ+p))A(x,χ(i∂φ+p))=R(χ(i∂φ)).(8) Note that the quantity J+J−corresponds to the HamiltonianH−(x,i∂φ+p)=A†(x,χ(i∂φ+p))A(x,χ(i∂φ+p)).(9)Identifyingi∂φJ3=−Hereξ(J3)≡−R(χ(i∂φ)).The algebraic shape invariance condition(3)is the main tool of our analysis.If we identifya0→χ(i∂φ);a1→χ(i∂φ+p),(12) then depending on the choice of theχfunction in eq.(12)we have:1.translational models:a1=a0+p forχ(z)=z,2.scaling models:a1=e p a0≡qa0forχ(z)=e z.One may note that in the translational reparametrization(χ(z)=z)the commutation rela-tions(11)describe a standard,undeformed SO(2,1)Lie algebra withξ(J3)=−2J3,as found in ref.[6].Other changes of parameters follow from more complicated choices forχ(z).For example, if one takesχ(z)=e e z,one gets the change of parameters a1=a20.Thus,with judicious choices forχ(z),various other reparametrizations can be emulated.In this letter,we will analyze the potential algebra of scaling type reparametrizations.Different models with such multiplicative reparametrizations are characterized by the form of the remainder function R(a0).As afirst example of scaling,we choose R(a0)=R1a0.This choice generates self-similar potentials studied in refs.[12,14].In this case,eqs.(11)become:[J3,J±]=±J±;[J+,J−]=ξ(J3)≡−R1exp(−pJ3).(13)Thus,we see that the resulting potential algebra is no longer given by a SO(2,1)Lie algebra,but is deformed.Tofind the energy spectrum of the Hamiltonian H−of eq.(9),wefirst construct the unitary representations of this deformed Lie algebra[15].We define,up to an additive constant, a function g(J3)such thatξ(J3)=g(J3)−g(J3−1).(14) The Casimir of this algebra is then given C2=J−J++g(J3).In a basis in which J3and C2are diagonal,J+and J−play the role of raising and lowering operators,respectively.Operating on an eigenstate|h of J3,we haveJ3|h =h|h ,J−|h =a(h)|h−1 ,J+|h =a⋆(h+1)|h+1 .(15)4Using eqs.(11)and (15)we obtain |a (h )|2−|a (h +1)|2=g (h )−g (h −1).(16)The profile of g (h )determines the dimension of the unitary representation.To illustrate how this mechanism works,let us consider the two cases presented in fig.1.One obtains finite(b)(a)Figure 1:Generic behaviors of g (h ).dimensional representations fig.1a,by starting from a point on the g (h )vs.h graph corresponding to h =h min ,and moving in integer steps parallel to the h -axis till the point corresponding to h =h max .Only those discrete values of h min are allowed for which at the end points a (h min )=a (h max +1)=0,and we get a finite representation.This is the case of SU (2)for example,where g (h )is given by the parabola h (h +1).For a monotonically decreasing function g (h ),fig.1b,there exists only one end point at h =h min .Starting from h min the value of h can be increased in integer steps till infinity.In this case we have an infinite dimensional representation.As in the finite case,h min labels the representation.However,the difference is that here h min takes continuous values.Similar arguments apply for a monotonically increasing function g (h ).A more detailed discussion of these techniques is given in ref.[15].In our case,from eqs.(13)and (14)one getsg (h )=R 11−q q −h ;q =e p ,(17)represented graphically in fig. 2.Note that for scaling problems [12],one requires 0<q <1,which leads to p <0.From the monotonically decreasing profile of the function g (h ),it follows that the unitary representations of this algebra are infinite dimensional.If we label the lowest weight state of the operator J 3by h min ,then a (h min )=0.Without loss of generality5we can choose the coefficients a(h)to be real.Then one obtains from(16)for an arbitrary h=h min+n,n=0,1,2,...q n−1a2(h)=g(h−n−1)−g(h−1)=R1q1−h|h .(19)q−1Therefore,the eigenenergy is:q n−1E n(h)=α(h)+W2(x,e i∂φ+p)−W′(x,e i∂φ+p)−E e ipφhψh min,n(x)=0,dx2or −d21h. . .Figure 2:Schematic diagram showing the correspondence between potential algebra and SUSYQM with scaling type reparametrization.For a more general case,we assume R (a 0)=∞j =1R j a j 0.In this caseg (h )=∞ j =1R j 1−q j ,(23)where αj (h )=R j e −j (h −1).This result agrees with those obtained in ref.[12].In conclusion,in this paper we have developed an algebraic approach involving nonlinear generalizations of Lie algebras for treating general shape invariant Hamiltonians.A.G.acknowledges a research leave and a grant from Loyola University Chicago which made his involvement in this work possible.R.D.would also like to thank the Physics Department of the University of Illinois at Chicago for warm hospitality.Partial financial support from the U.S.Department of Energy is gratefully acknowledged.7References[1]L.Infeld and T.E.Hull,Rev.Mod.Phys.23(1951)21.[2]L.E.Gendenshtein,JETP Lett.38(1983)356;L.E.Gendenshtein and I.V.Krive,Sov.p.28(1985)645.[3]F.Cooper,A.Khare,and U.Sukhatme,Phys.Rep.251(1995)268and references therein.[4]Y.Alhassid,F.G¨u rsey and F.Iachello,Ann.Phys.148(1983)346.[5]J.Wu and Y.Alhassid,Phys.Rev.A31(1990)557;H.Y.Cheung,Nuovo Cimento B101(1988)193.[6]A.Gangopadhyaya,J.V.Mallow and U.P.Sukhatme,Proceedings of Workshop on Super-symmetric Quantum Mechanics and Integrable Models,June1997,Editor:H.Aratyn et al., Springer-Verlag.[7]A.Gangopadhyaya,J.V.Mallow and U.P.Sukhatme,“Shape Invariant Natanzon Potentialsfrom Potential Algebra”;to appear in Phys.Rev.A.[8]A.B.Balantekin,Los Alamos e-print quant-ph/9712018.[9]A.Barut,A.Inomata and R.Wilson,Jour.Phys.A20(1987)4075and4083.[10]G.A.Natanzon,Teor.Mat.Fiz.,38(1979)219.[11]P.Cordero and S.Salamo,Foundations of Physics23(1993)675;Jour.Math.Phys.35(1994)3301.[12]D.Barclay,R.Dutt,A.Gangopadhyaya,A.Khare,A.Pagnamenta and U.Sukhatme,Phys.Rev.A48(1993)2786.[13]J.Beckers,Y.Brihaye,and N.Debergh,Los Alamos preprint hep-th/9803253[14]V.P.Spiridonov,Phys.Rev.Lett.69(1992)298.[15]M.Rocek,Phys.Lett.B255(1991)554.8。

维纳辛坎定理全文共四篇示例，供读者参考第一篇示例：维纳辛坎定理（Winzerkind theorem）是一个重要的数学定理，它描述了一种关于离散嵌入维数的方法。

它是由德国数学家维纳辛坎（Winzerkind）在1923年发现的。

在数学领域中，维纳辛坎定理被广泛应用于研究网络、图像处理、数据挖掘等问题。

维纳辛坎定理的核心思想是，任意一个离散数据集都可以通过一个嵌入函数映射到一个低维空间中，而且这个低维空间的维数与原始数据集保持一致。

换句话说，通过维纳辛坎定理，我们可以将高维数据转换为低维数据，并且不会损失任何信息。

在实际应用中，维纳辛坎定理可以用来解决维数灾难（curse of dimensionality）的问题。

维数灾难指的是当数据的维数很高时，数据分布变得稀疏，计算复杂度急剧增加，导致传统的学习算法效果变差。

通过维纳辛坎定理，我们可以将高维数据映射到低维空间中，从而在数据处理和学习算法上获得更好的效果。

除了在机器学习和数据分析领域中的应用，维纳辛坎定理还被广泛用于图像处理、模式识别、生物信息学等领域。

在图像处理中，我们可以利用维纳辛坎定理将高分辨率的图像转换为低分辨率图像，以便于计算和存储。

在生物信息学中，我们可以使用维纳辛坎定理来降低基因组数据的复杂度，从而更好地理解基因间的关系。

维纳辛坎定理是一个非常重要且有实际应用的数学定理。

通过将高维数据映射到低维空间中，我们可以更好地理解数据的结构和特征，从而为数据处理和分析提供更有效的方法。

希望本文能够帮助读者更深入地了解维纳辛坎定理及其在现实生活中的应用。

第二篇示例：维纳辛坎定理（Weierstrass-Casorati Theorem）是分析数学中的一个重要定理，它揭示了复变函数的性质和奇点行为。

该定理由德国数学家卡尔·魏尔斯特拉斯（Karl Weierstrass）和意大利数学家费利切·卡索拉蒂（Felice Casorati）独立证明，并于19世纪中叶首次发布。

美国数学竞赛amc8的常用数学英语单词等于equals, is equal to, is equivalent to 大于is greater than 小于is lesser than 大于等于is equal or greater than 小于等于is equal or lesser than运算符operator 数字digit 数number 自然数natural number公理axiom 定理theorem 计算calculation 运算operation 算术arithmetic 证明prove 假设hypothesis, hypotheses（pl.）命题proposition加plus（prep.）, add（v.）, addition（n.）被加数augend, summand 加数addend 和sum减minus（prep.）, subtract（v.）, subtraction（n.）被减数minuend 减数subtrahend 差remainder乘times（prep.）, multiply（v.）, multiplication（n.被乘数multiplicand, faciend 乘数multiplicator 积product除divided by（prep.）, divide（v.）, division（n.）被除数dividend 除数divisor 商quotient整数integer 小数decimal 小数点decimal point分数fraction 分子numerator 分母denominator 比ratio正positive 负negative 零null, zero, nought, nil十进制decimal system 二进制binary system 十六进制hexadecimal system权weight, significance 进位carry 截尾truncation四舍五入round 下舍入round down 上舍入round up有效数字significant digit 无效数字insignificant digit代数algebra 公式formula, formulae（pl.）单项式monomial 多项式polynomial, multinomial 系数coefficient未知数unknown, x-factor, y-factor, z-factor等式，方程式equation 一次方程simple equation 二次方程quadratic equation 不等式inequation 三次方程cubic equation 四次方程quartic equation阶乘factorial 对数logarithm 指数，幂exponent乘方power 二次方，平方square 三次方，立方cube四次方the power of four, the fourth powern次方the power of n, the nth power开方evolution, extraction 二次方根，平方根square root三次方根，立方根cube root 四次方根the root of four, the fourth rootn次方根the root of n, the nth root集合aggregate 元素element 空集void 子集subset 交集intersection 并集union 补集complement映射mapping 图象image 函数function 定义域domain, field of definition 值域range 周期性periodicity常量constant 变量variable 单调性monotonicity 奇偶性parity数列，级数series 微积分calculus 微分differential 导数derivative积分integral 定积分definite integral 不定积分indefinite integral有理数rational number 无理数irrational number 实数real number虚数imaginary number 复数complex number极限limit 无穷大infinite（a.）infinity（n.）无穷小infinitesimal矩阵matrix 行列式determinant几何geometry 点point 线line 面plane 体solid 线段segment射线radial 平行parallel 相交intersect角angle 角度degree 弧度radian 周角perigon锐角acute angle 直角right angle 钝角obtuse angle 平角straight angle底base 边side 高height直角边leg 斜边hypotenuse 勾股定理Pythagorean theorem三角形triangle 锐角三角形acute triangle 直角三角形right triangle钝角三角形obtuse triangle 不等边三角形scalene triangle等腰三角形isosceles triangle 等边三角形equilateral triangle长length 宽width四边形quadrilateral 平行四边形parallelogram 矩形rectangle菱形rhomb, rhombus, rhombi（pl.）, diamond 正方形square梯形trapezoid 直角梯形right trapezoid 等腰梯形isosceles trapezoid多边形polygon 正多边形equilateral polygon 五边形pentagon六边形hexagon 七边形heptagon 八边形octagon 九边形enneagon十边形decagon 十一边形hendecagon 十二边形dodecagon圆circle 圆心centre（BrE）, center（AmE）半径radius 直径diameter 圆周率pi 弧arc 半圆semicircle扇形sector 环ring 椭圆ellipse圆周circumference 周长perimeter 面积area 轨迹locus, loca（pl.）相似similar 全等congruent立方体cube 多面体polyhedron 四面体tetrahedron 五面体pentahedron 六面体hexahedron 平行六面体parallelepiped 七面体heptahedron八面体octahedron 九面体enneahedron 十面体decahedron十一面体hendecahedron 十二面体dodecahedron 二十面体icosahedron棱锥pyramid 棱柱prism 棱台frustum of a prism旋转rotation 轴axis 圆锥cone 圆柱cylinder 圆台frustum of a cone球sphere 半球hemisphere底面undersurface 表面积surface area 体积volume空间space 坐标系coordinates 坐标轴x-axis, y-axis, z-axis 原点origin横坐标x-coordinate 纵坐标y-coordinate 双曲线hyperbola 抛物线parabola三角trigonometry 正弦sine 余弦cosine 反正弦arc sine 反余弦arc cosine 正切tangent 余切cotangent 反正切arc tangent 反余切arc cotangent正割secant 余割cosecant 反正割arc secant 反余割arc cosecant相位phase 周期period 振幅amplitude内切圆inscribed circle 外切圆circumcircle内心incentre（BrE）, incenter（AmE）外心excentre（BrE）, excenter（AmE）旁心escentre（BrE）, escenter（AmE）垂心orthocentre（BrE）, orthocenter（AmE）重心barycentre（BrE）, barycenter（AmE）统计statistics 平均数average 加权平均数weighted average概率，或然率probability标准差root-mean-square deviation, standard deviation 分布distribution 正态分布normal distribution 非正态分布abnormal distribution图表graph 条形统计图bar graph 柱形统计图histogram折线统计图broken line graph 曲线统计图curve diagram扇形统计图pie diagram比例propotion 百分比percent 百分点percentage 百分位数percentile排列permutation 组合combination。

Unit1 mathematics名词解释绝对补集absolute complement / 代数algebra /代数式algebraic expression / 代数方程algebraic equation / 代数不等式algebraic inequality / 任意常数arbitrary constant / 数组array / 底数；基数base number / 连续函数continuous function / 函数function / 复合函数function of function / 函数记号functional notation / 集合aggregate / 子集subset /迭代函数iterative function/优先权之争priority battle/分形特征fractal properties/有意义make sense/以越来越小的规模重复同一模式patterns repeat themselves at smaller and smaller scales/混沌理论chaos theory/季刊a quarterly journal/数学界the mathematics community/波纹线crisp lines/会议公报proceedings of a conference翻译3. Translate the sentences into Chinese.1)他主要是因为用分形这个概念来描述（海岸线、雪花、山脉和树木）等不规则形状等现象而闻名于世，这些不规则形状在越来越小的规模上不断重复同一模式。

2)如果再仔细观察，就可以发现集的边界并没有呈波纹线，而是像火焰一样闪光。

3)但是，克朗兹在这场辩论中引入了一个新东西，他说曼德布洛特集不是曼德布洛特集发明的，而是早在“曼德布洛特集”这个术语出现几年以前就已经明确地在数学文献中出现了。

4)曼德布洛特同时也暗示即使布鲁克斯和马特尔斯基的论文先于他发表，但因为他们没有领会到其价值，仍然不能将他们看作是曼德布洛特集的发现者。

(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。

0+||zero-dagger; 读作零正。

1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。

AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。

BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。

B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。

C0 类函数||function of class C0; 又称“连续函数类”。

CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。

Cp统计量||Cp-statisticC。

以下是90个超新星纪元的词语以及它们的译文：1.超新星纪元- Supernova Age2.星际物质- Interstellar Matter3.星云- Nebula4.恒星- Star5.黑洞- Black Hole6.脉冲星- Pulsar7.星系- Galaxy8.旋臂- Spiral Arm9.恒星系- Stellar System10.双星- Binary Star11.星团- Star Cluster12.星云团- Nebula Cluster13.星际空间- Interstellar Space14.星尘- Stardust15.星系团- Galaxy Cluster16.超星系团- Supercluster17.宇宙射线- Cosmic Ray18.反物质- Antimatter19.高能射线- High-energy Radiation20.量子力学- Quantum Mechanics21.引力波- Gravitational Wave22.宇宙微波背景辐射- Cosmic Microwave Background Radiation23.暗物质- Dark Matter24.暗能量- Dark Energy25.星系核- Galaxy Core26.黑洞吞噬- Black Hole Accretion27.恒星演化- Stellar Evolution28.核合成- Nucleosynthesis29.星系碰撞- Galaxy Collision30.恒星爆炸- Stellar Explosion31.引力透镜- Gravitational Lensing32.宇宙网- Cosmic Web33.反物质粒子- Antimatter Particles34.高纬宇宙模型- High-dimensional Cosmic Model35.宇宙常数- Cosmological Constant36.宇宙密度- Cosmic Density37.宇宙膨胀- Cosmic Expansion38.宇宙学红移- Cosmological Redshift39.大爆炸理论- Big Bang Theory40.弦理论- String Theory41.相对论- Relativity42.量子力学- Quantum Mechanics43.弦理论- String Theory44.卡鲁扎-克莱因理论- Kaluza-Klein Theory45.高维时空- Higher-dimensional Spacetime46.虚时间- Virtual Time47.宇宙微波背景辐射- Cosmic Microwave Background Radiation48.标准宇宙模型- Standard Cosmological Model49.星系团- Galaxy Cluster50.超星系团- Supercluster51.丝状结构- Filamentary Structure52.大尺度结构- Large-scale Structure53.哈勃常数- Hubble Constant54.引力波- Gravitational Wave55.黑洞信息悖论- Black Hole Information Paradox56.夸克星- Quark Star57.反物质星- Antimatter Star58.原子核- Atomic Nucleus59.量子纠缠- Quantum Entanglement60.高温高压状态方程- High-temperature and high-pressure equation ofstate61.星系演化- Galaxy Evolution62.星系动力学- Galaxy Dynamics63.星际物质循环- Interstellar Matter Cycle64.恒星形成- Star Formation65.分子云- Molecular Cloud66.星际空间气体- Interstellar Gas67.星际尘埃- Interstellar Dust68.星系核活动- Active Galactic Nucleus69.射电星系- Radio Galaxy70.光学星系- Optical Galaxy71.X射线星系- X-ray Galaxy72.恒星团- Star Cluster73.双星系统- Binary Star System74.变星- Variable Star75.新星- Nova76.超新星- Supernova77.中子星- Neutron Star78.脉冲星- Pulsar79.黑洞候选体- Black Hole Candidate80.高光度蓝变星- High-luminosity Blue Variable Star81.超巨星- Supergiant Star82.红巨星- Red Giant Star83.黄矮星- Yellow Dwarf Star84.白矮星- White Dwarf Star85.恒星演化模型- Stellar Evolution Model86.星际物质分布- Interstellar Matter Distribution87.分子光谱学- Molecular Spectroscopy88.高能天体物理学- High-energy Astrophysics89.天体化学- Astrochemistry90.宇宙射线物理学- Cosmic Ray Physics。

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庞加莱猜想-前言Wir m\"ussen wissen! Wir werden wissen!(我们必须知道！我们必将知道！)—— David Hilbert两年前科学版举行过一次版聚，我报告了低维拓扑里面的一些问题和进展，其中有一半篇幅是关于Poincar\'e 猜想。

版聚后，flyleaf 要求大家回去后把自己所讲的内容发在版上。

当时我甚至已经开始写了一两段，但后来又搁置了。

主要是因为自己对于低维拓扑还是一个门外汉，写出来的东西难免有疏漏之处，不敢妄下笔。

两年过去，我对低维拓扑这门学科的了解比原先多了，说话的底气也就比原先足了。

另外，由于Clay 研究所的百万巨赏，近年来Poincar\'e 猜想频频在媒体上曝光；而且Perelman 最近的工作使数学家们有理由相信我们已经充分接近于这一猜想的最后解决。

所以大概会有很多人对Poincar\'e 猜想的来龙去脉感兴趣，我也好借机一偿两年来的宿愿。

现代科学的高速发展使各学科之间的鸿沟加大，不同学科之间难以互相理解，所以非数学专业的读者在阅读本文时可能会遇到一些困难。

但限于篇幅和文章的形式，我也不可能对很多东西详细解释。

一些最基本的拓扑概念如“流形”，我将在本文的附录中解释。

还有一些“同调群”、“基本群”之类的名词，读者见到时大可不去理会它们的确切含义。

我将尽量避免使用这一类的专业术语。

作者并非拓扑方面的专家，对下面要说的很多内容都是道听途说，只知其然而不知其所以然；作者更不善于写作，写出来的东东总会枯燥无味，难登大雅之堂。

凡此种种，还请读者诸君海涵。

问题的由来Consid\'erons maintenant une vari\'et\'e [ferm\'ee] $V$ \`a trois dimensions ... Est-il possible que le groupe fondamental de $V$ ser\'eduise \`a la substitution identique, et que pourtant $V$ ne soit pas simplement connexe?—— Henri Poincar\'e在拓扑学家的眼里，篮球、排球和乒乓球并没有什么不同，它们都同胚于三维空间中的球面S^2. (我们把n+1维欧氏空间中到原点距离为1的点的集合记作S^n，称为n维球面(sphere)。