Articles

AN UNGUIDED TOUR STARTED FROM CHIRALITY

  • JIANG Ba-Ju ,
  • WANG Shi-Cheng
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  • Institute of Mathematical Science, Peking University, Beijing 100871, China

Received date: 2008-12-31

  Online published: 2009-07-20

Abstract

This survey article records an unguided mathematical tour by topologists at Peking University and their collaborators in the last ten years. The tour started from research on chirality and, attracted by questions around attractors, led to a zigzag path across topology and dynamics.
People who joined us in this tour at various stages include Ding Fan, Liu Yi, Ni Yi, Pan Jianzhong, Yao Jiangang, Zheng Hao and Zhou Qing. Conversations with Robert Edwards, Wen Lan and others, added to the twists and turns that made the trip more fun. This article benefits from related lectures by these authors, in many conferences, universi-ties, as well as high schools.

Cite this article

JIANG Ba-Ju , WANG Shi-Cheng . AN UNGUIDED TOUR STARTED FROM CHIRALITY[J]. Acta mathematica scientia, Series B, 2009 , 29(4) : 867 -880 . DOI: 10.1016/S0252-9602(09)60075-8

References


[1] Bing R H. A simple closed curve is the only homogeneous bounded plane continuum that contains an arc.
Canad J Math, 1960, 12: 209–230


[2] Bothe H G. The ambient structure of expanding attractors II. Solenoids in 3-manifolds. Math Nachr, 1983,
112: 69–102


[3] Chevalley D, Eilenberg S. Cohomology theory of Lie groups and Lie algebras. Trans Amer Math Soc, 1948, 63: 85–124


[4] Ding F, Liu Y, Wang S C, Yao J G. Extending Tp automorphisms over Rp+2 and realizing DE attractors.
math.GT (math.DS). arXiv:0811.4032


[5] Ding F, Pan J Z, Wang S C, Yao J G. Manifolds with (f) a union of DE attractors are rational homology
spheres. math.GT (math.DS). arXiv:0812.1260


[6] Epstein D, Shub M. Expanding endomorphisms of flat manifolds. Topology, 1968, 7: 139–141


[7] Freedman B, Freedman M H. Kneser-Haken finiteness for bounded 3-manifolds, locally free groups, and
cyclic covers. Topology, 1998, 37: 133–147


[8] Franks J, Williams B. Anomalous Anosov flows//Global Theory of Dynamical Systems. Lecture Notes in
Math 819. Berlin: Springer, 1980: 158–174


[9] Gordon C. On embedding infinite cyclic covers of knot spaces into compact 3-manifold. Math.GT/0608339


[10] Gabai D. Foliations and the topology of 3-manifolds III. J Differ Geom, 1987, 26: 479–536


[11] Gromov M. Groups of polynomial growth and expanding maps. Inst Hautes Tudes Sci Publ Math, 1981,
53: 53–73


[12] Hirose S. On diffeomorphisms over surfaces trivially embedded in the 4-sphere. Algebr Geom Topol, 2002,
2: 791–824


[13] Jiang B J, Lin X L, Wang S C, Wu Y Q. Achirality of knots and links. Topology Appl, 2002, 119(2): 185–208


[14] Jiang B J, Wang S C. Achirality and planarity. Commun Contemp Math, 2000, 2(3): 299–305


[15] Jiang B J, Ni Y, Wang S C. 3-manifolds that admit knotted solenoids as attractors. Trans Amer Math Soc, 2004, 356(11): 4371–4382


[16] Jiang B J, Ni Y,Wang S C, Zhou Q. Embedding infinite cyclic covers of knot spaces into 3-space. Topology,
2006, 45(4): 691–705


[17] Jiang B J, Wang S C, Zhou Q, Zheng H. On tame embeddings of solenoids into 3-space. mah.GT.
arXiv:math/0611900v1


[18] Jiang B J, Wang S C, Zheng H. No embeddings of solenoids into surfaces. Proc Amer Math Soc, 2008,
136(10): 3697–3700


[19] Kirby R C. The Topology of 4-Manifolds. Lecture Notes in Math 1374. Berlin: Springer-Verlag, 1990


[20] Ma J, Yu B. The realization of Smale solenoid type attractors in 3-manifolds. Topology Appl, 2007, 154(17): 3021–3031


[21] Ma J, Yu B. Genus two Smale-Williams solenoids as attractors in 3-manifolds.


[22] McCord M C. Inverse limit sequences with covering maps. Trans Amer Math Soc, 1965, 114(1): 197–209


[23] Montesinos J M. On twins in the four-sphere I. Quart J Math, 1983, 34(2): 171–199


[24] Nomizu K. On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann Math, 1954,
59: 531–538


[25] Robinson C. Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. 2nd ed. Studies in Advanced
Mathematics. Boca Raton, FL: CRC Press, 1999


[26] Shub M. Expanding maps//Global Analysis: Proc Sympos Pure Math, Vol 14. Amer Math Soc, 1970: 273–276


[27] Simon J. Topological chirality of certain molecules. Topology, 1986, 25(2): 229–235


[28] Smale S. Differentiable dynamical systems. Bull Amer Math Soc, 1967, 73: 747–817


[29] Soma T. The Gromov invariant of links. Invent Math, 1981, 64: 445–454


[30] Vietoris L. Über den höheren Zusammenhang kompakter R¨aume und eine Klasse von zusammenhangstreuen Abbildungen. Math Ann, 1927, 97(1): 454–472


[31] Wang S D. Strict achirality of prime links up to 11-crossing. Acta Math Sin (Engl Ser), 2008, 24(6): 997–1004


[32] Yang Z Q. On embedding infinite cyclic covers of knot spaces into 3-space.

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