×

Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere. (English) Zbl 1101.57306

Summary: We present a computer program based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices. As an example, we obtain a 16-vertex triangulation of the Poincaré homology 3-sphere; we construct an infinite series of non-PL \(d\)-dimensional spheres with \(d+13\) vertices for \(d\geq 5\); and we show that if a \(d\)-manifold, with \(d\geq 5\), admits any triangulation on \(n\) vertices, it admits a noncombinatorial triangulation on \(n+12\) vertices.

MSC:

57Q15 Triangulating manifolds
57-04 Software, source code, etc. for problems pertaining to manifolds and cell complexes
57M15 Relations of low-dimensional topology with graph theory
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)

Software:

GAP; BISTELLAR
PDF BibTeX XML Cite
Full Text: DOI Euclid EuDML

References:

[1] Akbulut S., Casson’s invariant for oriented homology 3-spheres: an exposition (1990) · Zbl 0695.57011
[2] Altshuler A., Discrete Math. 8 pp 113– (1974) · Zbl 0292.57011
[3] Altshuler A., Discrete Math. 16 (2) pp 91– (1976) · Zbl 0348.57002
[4] Bagchi B., Discrete Math. 188 (1) pp 41– (1998) · Zbl 0951.57012
[5] Barnette D., Discrete Math. 16 (4) pp 291– (1976) · Zbl 0345.57007
[6] Billera L. J., Handbook of discrete and computational geometry pp 291– (1997)
[7] Billera L. J., J. Combin. Theory Ser. A 31 (3) pp 237– (1981) · Zbl 0479.52006
[8] Bredon G. E., Introduction to compact transformation groups (1972) · Zbl 0246.57017
[9] Brehm U., Topology 26 (4) pp 465– (1987) · Zbl 0681.57009
[10] Brehm U., ”Triangulations of lens spaces with few simplices” (1993)
[11] Cannon J. W., Ann. of Math. (2) 110 (1) pp 83– (1979) · Zbl 0424.57007
[12] Daverman R. J., Decompositions of manifolds (1986) · Zbl 0608.57002
[13] Dehn M., Math. Ann. 69 pp 137– (1910) · JFM 41.0543.01
[14] Edwards R. D., Notices Amer. Math. Soc. 22 pp A–334– (1975)
[15] Freedman M. H., Selected applications of geometry to low-dimensional topology (1989)
[16] Freedman M. H., Topology of 4-manifolds (1990) · Zbl 0705.57001
[17] Glaser L. C., Geometrical combinatorial topology 1 (1970) · Zbl 0212.55603
[18] Heckenbach F., HOMOLOGY (computer program) and Die Möbiusfunktion und Homologien auf partiell geordneten Mengen (1997)
[19] Hudson J. F. P., Piecewise linear topology (1969) · Zbl 0189.54507
[20] Kirby R. C., Geometric topology (Athens, GA, 1977) pp 113– (1979)
[21] Kirby R. C., Foundational essays on topological manifolds, smoothings, and triangulations (1977) · Zbl 0361.57004
[22] DOI: 10.1126/science.220.4598.671 · Zbl 1225.90162
[23] Klee V., Canad. J. Math. 16 pp 517– (1964) · Zbl 0134.42403
[24] Kneser H., Jahresb. Deutsch. Math. Verein. 38 pp 248– (1929)
[25] Köhler E., ”Combinatorial manifolds with transitive automorphism group on few vertices” (1999)
[26] Kühnel W., Advances in differential geometry and topology pp 59– (1990)
[27] Kühnel W., Tight polyhedral submanifolds and tight triangulations (1995) · Zbl 0834.53004
[28] Kühnel W., Math. Intelligencer 5 (3) pp 11– (1983) · Zbl 0534.51009
[29] Kühnel W., ”A census of tight triangulations” (1999) · Zbl 0996.52014
[30] Kuiper, N. H. ”A short history of triangulation and related matters”. Proc. Bicentennial Congress Wiskundig Genootschap. 1978, Amsterdam. vol. 1, pp.61–79. Amsterdam: Math. Centre. [Kuiper 1979], Math. Centre Tracts 100
[31] Lashof R., Ann. of Math. (2) 81 pp 565– (1965) · Zbl 0137.17601
[32] Lickorish W. B. R., Geometric topology (Athens, GA, 1993) pp 375– (1997)
[33] Lutz F. H., ”BISTELLAR, second version, 02/99” (1999)
[34] Lutz F. H., ”BISTELLAR_EQUIVALENT, version 02/99” (1999)
[35] Lutz F. H., ”Examples of Z-acyclic and contractible vertex-homogeneous simplicial complexes” (1999)
[36] Lutz F. H., Triangulated manifolds with few vertices and vertex-transitive group actions (1999) · Zbl 0977.57030
[37] Marin, A. 1988.”Un nouvel invariant pour les sphéres d’homologie de dimension trois (d’aprés Casson)”151–164. [Marin 1988], Exp. No. 693 inSéminaire Bourbaki, 1987/88, Asterisque 161–162
[38] DOI: 10.1080/10586458.1998.10504365 · Zbl 0916.57017
[39] DOI: 10.2307/1969769 · Zbl 0048.17102
[40] Moise, E. E. 1977.Geometric topology in dimensions 2and3, Graduate Texts in Math47New York: Springer. [Moise 1977]
[41] Nabutovsky A., Comm. Math. Phys. 181 (2) pp 303– (1996) · Zbl 0863.57015
[42] DOI: 10.1007/BF02941601 · Zbl 0651.52007
[43] Poincaré H., Rend. Circ. Mat. Palermo 18 pp 45– (1904) · JFM 35.0504.13
[44] Rado T., Acta Litt. Sci. Szeged 2 pp 101– (1925)
[45] Rinnooy Kan A. H. G., Handbook in operations research and management science, vol. 1: Optimization pp 631– (1989)
[46] Rolfsen D., Knots and links (1976) · Zbl 0339.55004
[47] Rourke C. P., Introduction to piecewise-linear topology (1972) · Zbl 0254.57010
[48] Schonert M., GAP: Groups, algorithms, and programming,, 5. ed. (1996)
[49] Stanley R. P., Adv. in Math. 35 (3) pp 236– (1980) · Zbl 0427.52006
[50] Stanley R. P., Discrete geometry and convexity (New York, 1982) pp 212– (1985)
[51] DOI: 10.1007/978-1-4612-4372-4
[52] Thompson A., Math. Res. Lett. 1 (5) pp 613– (1994) · Zbl 0849.57009
[53] DOI: 10.1007/BF01457920 · Zbl 0006.03403
[54] DOI: 10.1007/BF02392331 · Zbl 0204.56301
[55] Weber C., Math. Z. 37 pp 237– (1933) · Zbl 0007.02806
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.