Alternating Heegaard diagrams and Williams solenoid attractors in $3$-manifolds

Chao Wang, Yimu Zhang

DOI: http://dx.doi.org/10.12775/TMNA.2016.033

Abstract


We find all Heegaard diagrams with the property ``alternating'' or ``weakly alternating'' on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two $3$-manifolds, each admits an automorphism whose non-wandering set consists of two Williams solenoids, one attractor and one repeller. These manifolds contain half of Prism manifolds, Poincaré's homology $3$-sphere and many other Seifert manifolds, all integer Dehn surgeries on the figure eight knot, also many connected sums. The result shows that many kinds of $3$-manifolds admit a kind of ``translation'' with certain stability.

Keywords


Heegaard diagram; solenoid attractor; Prism manifold; Poincaré's homology $3$-sphere; figure eight knot

Full Text:

PREVIEW FULL TEXT

References


M. Boileau, S. Maillot and J. Porti, Three-dimensional orbifolds and their geometric structures, Panoramas et Synthèses 15. Société Mathḿatique de France, Paris 2003.

G. Burde and H. Zieschang, Knots, de Gruyter Stud. Math. 5, Walter de Gruyter, Berlin, New York, 1985.

B. Jiang and Y. Ni, S. Wang, 3-manifolds that admit knotted solenoids as attractors, Trans. Amer. Math. Soc. 356 (2004), 4371–4382.

M. Montesinos, Classical tessellations and three-manifolds, Springer–Verlag, 1985.

J. Ma and B. Yu, Genus two Smale–Williams solenoid attractors in 3-manifolds, J. Knot Theory Ramifications 20 (2011), 909–926.

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

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817.

W.P. Thurston, The geometry and topology of three-manifolds, Lecture Notes, 1978.

R.F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473–487.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism