Combinatorial 3-manifolds with transitive cyclic symmetry.

Authors
BookarXiv.org.
Year of edition2011

 In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices. Finally, a combination of these results is used to describe new infinite families of transitive cyclic combinatorial manifolds and in particular a family of neighborly combinatorial lens spaces of infinitely many distinct topological types.

 24 pages, 5 figures. Journal-ref: Discrete and Computational Geometry, 51(2):394-426, 2014

Reference on publication
Spreer J.   Combinatorial 3-manifolds with transitive cyclic symmetry. - : , 2011. // arXiv.org, 2011.
Bibliography
1.3 A. Altshuler. Polyhedral realization in R of triangulations of the torus and 2-manifolds in cyclic 4-polytopes. Discrete Math., 1(3):211--238, 1971/1972.
2.A. Altshuler and L. Steinberg. Neighborly combinatorial 3-manifolds with 9 vertices. Discrete Math., 8:113--137, 1974.
3.T. Banchoff. Critical points and curvature for embedded polyhedra. J. Differential Geom., 1:245--256, 1967.
4.T. F. Banchoff. Critical points and curvature for embedded polyhedra. II. In Differential geometry (College Park, Md., 1981/1982), volume 32 of Progr. Math., pages 34--55. Birkh?auser Boston, Boston, MA, 1983.
5.L. Bieberbach. ?Uber die Bewegungsgruppen der Euklidischen R?aume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich. Math. Ann., 72(3):400--412, 1912.
6.A. Bj?orner and F. H. Lutz. Simplicial manifolds, bistellar flips and a 16vertex triangulation of the Poincar?e homology 3-sphere. Experiment. Math., 9(2):275--289, 2000.
7.3 U. Brehm and W. K?uhnel. Lattice triangulations of E and of the 3-torus. Israel J. Math., 189:97--133, 2012.
8.B. A. Burton. Enumeration of non-orientable 3-manifolds using face-pairing graphs and union-find. Discrete Comput. Geom., 38(3):527--571, 2007.
9.B. A. Burton, R. Budney, W. Pettersson, et al. Regina: normal surface and 3-manifold topology software, version 4.93. http://regina.sourceforge. net/, 1999--2012.
10.B. A. Burton and J. Spreer. Combinatorial Seifert Fibred spaces with transitive cyclic automorphism group, 2013. In preparation, 23 pages, 7 figures.
11.F. Effenberger and J. Spreer. simpcomp - a GAP toolbox for simplicial complexes. ACM Communications in Computer Algebra, 44(4):186 -- 189, 2010.
12.F. Effenberger and J. Spreer. simpcomp - a GAP package, Version 1.6.1. http://www.igt.uni-stuttgart.de/LstDiffgeo/simpcomp, 2013.
13.F. Effenberger and J. Spreer. Simplicial blowups and discrete normal surfaces in the GAP package simpcomp. ACM Communications in Computer Algebra, 45(3):173 -- 176, 2011.
14.A. Emch. Triple and multiple systems, their geometric configurations and groups. Trans. Amer. Math. Soc., 31(1):25--42, 1929.
15.GAP-- Groups, Algorithms, and Programming, Version 4.5.6. http://www.gap-system.org, 2012.
16.J. Hempel. 3-Manifolds. Annals of Mathematics Studies, pages 24--26, 1976.
17.W. K?uhnel. Tight polyhedral submanifolds and tight triangulations, volume 1612 of Lecture Notes in Math. Springer, 1995.
18.W. K?uhnel. Centrally-symmetric tight surfaces and graph embeddings. Beitr?age Algebra Geom., 37(2):347--354, 1996.
19.W. K?uhnel. Topological aspects of twofold triple systems. Exposition. Math., 16(4):289--332, 1998.
20.W. K?uhnel. Differential geometry, volume 16 of Student Mathematical Library. American Mathematical Society, Providence, RI, 2002. Curves--surfaces---manifolds, Translated from the 1999 German original by Bruce Hunt.
21.W. K?uhnel and G. Lassmann. Neighborly combinatorial 3-manifolds with dihedral automorphism group. Israel J. Math., 52(1-2):147--166, 1985.
22.W. K?uhnel and G. Lassmann. Combinatorial d-tori with a large symmetry group. Discrete Comput. Geom., 3(2):169--176, 1988.
23.W. K?uhnel and G. Lassmann. Permuted difference cycles and triangulated sphere bundles. Discrete Math., 162(1-3):215--227, 1996.
24.N. H. Kuiper. Morse relations for curvature and tightness. In Proceedings of Liverpool Singularities Symposium, II (1969/1970), volume 209 of Lecture Notes in Math., pages 77--89, Berlin, 1971.
25.C. C. Lindner and A. Rosa, editors. Topics on Steiner systems, volume 7 of Ann. Discrete Math. North-Holland Publishing Co., Amsterdam, 1980.
26.E. Luft and D. Sjerve. 3-manifolds with subgroups Z ? Z ? Z in their fundamental groups. Pacific J. Math., 114(1):191--205, 1984.
27.F. H. Lutz. The Manifold Page. http://page.math.tu-berlin.de/ lutz/stellar/. ~
28.F. H. Lutz. Triangulating manifolds. In press, ISBN 978-3-540-34502-2.
29.F. H. Lutz. Triangulated manifolds with few vertices and vertex-transitive group actions. PhD thesis, TU Berlin, Aachen, 1999.
30.F. H. Lutz. Equivelar and d-covered triangulations of surfaces. II. Cyclic triangulations and tessellations. arXiv:1001.2779v1 [math.CO], 2010. To appear in Contrib. Discr. Math.
31.F. H. Lutz, T. Sulanke, and E. Swartz. f-vectors of 3-manifolds. Electron. J. Comb., 16(2):Research Paper 13, 33, 2009.
32.S. Matveev et al. Three-manifold Recognizer. http://matlas.math.csu.ru/, 2013.
33.J. Milnor. Towards the Poincar?e conjecture and the classification of 3manifolds. Notices Amer. Math. Soc., 50(10):1226--1233, 2003.
34.P. Orlik. Seifert manifolds. Lecture Notes in Mathematics, Vol. 291. Springer, 1972.
35.G. Perelman. The entropy formula for the ricci flow and its geometric applications. arXiv:math.DG/0211159, 2002.
36.G. Perelman. Finite extinction time for the solutions to the ricci flow on certain three-manifolds. arXiv:math.DG/0307245, 2003.
37.G. Perelman. Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109, 2003.
38.G. Ringel. Map color theorem, volume 209 of Die Grundlehren der mathematischen Wissenschaften. Springer, 1974.
39.J. H. Rubinstein. An algorithm to recognize the 3-sphere. In Proceedings of the International Congress of Mathematicians (Z?urich, 1994), volume 1, pages 601--611. Birkh?auser, 1995.
40.N. Saveliev. Invariants of Homology 3-Spheres, volume 1 of Low-dimensional topology, Encyclopaedia of mathematical sciences. Springer, 2002.
41.J. Spreer. Blowups, slicings and permutation groups in combinatorial topology. Logos Verlag Berlin, 2011.
42.J. Spreer. Normal surfaces as combinatorial slicings. Discrete Math., 311(14):1295--1309, 2011. doi:10.1016/j.disc.2011.03.013.
43.J. R. Stallings. Some topological proofs and extensions of Grushko's theorem. PhD thesis, Princeton University, 1959.
44.W. Threlfall and H. Seifert. Topologische Untersuchung der Diskontinuit?atsbereiche endlicher Bewegungsgruppen des dreidimensionalen sph?arischen Raumes, Schlu ?. Math. Ann., 107:543--586, 1933.
45.W. P. Thurston. The geometry and topology of 3-manifolds, volume 1. Princeton University Press, Princeton, N.J., 1980. Electronic version 1.1 March 2002.
46.V. G. Turaev and O. Y. Viro. State sum invariants of 3-manifolds and quantum 6j-symbols. Topology, 31(4):865--902, 1992.
47.J. Weeks. SnapPy (Software for hyperbolic 3-manifolds). http://www.math.uic.edu/t3m/SnapPy/, 1999.

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