Brieskorn manifolds, generated Sieradski groups, and coverings of lens space

A. Yu. Vesnin ; T. A. Kozlovskaya

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 85-97

Résumé

The Brieskorn manifold $\mathscr B(p,q,r)$ is the $r$-fold cyclic covering of the three-dimensional sphere $S^{3}$ branched over the torus knot $T(p,q)$. The generalised Sieradski groups $S(m,p,q)$ are groups with $m$-cyclic presentation $G_{m}(w)$, where the word $w$ has a special form depending on $p$ and $q$. In particular, $S(m,3,2)=G_{m}(w)$ is the group with $m$ generators $x_{1},\ldots,x_{m}$ and $m$ defining relations $w(x_{i}, x_{i+1}, x_{i+2})=1$, where $w(x_{i}, x_{i+1}, x_{i+2}) = x_{i} x_{i+2} x_{i+1}^{-1}$. Cyclic presentations of $S(2n,3,2)$ in the form $G_{n}(w)$ were investigated by Howie and Williams, who showed that the $n$-cyclic presentations are geometric, i.e., correspond to the spines of closed three-dimensional manifolds. We establish an analogous result for the groups $S(2n,5,2)$. It is shown that in both cases the manifolds are $n$-fold branched cyclic coverings of lens spaces. For the classification of the constructed manifolds, we use Matveev's computer program “Recognizer.”

Détail

Keywords: three-dimensional manifold, Brieskorn manifold, cyclically presented group, Sieradski group, branched covering.
Mots-clés : lens space

Bibliographie

[1] Brieskorn E., “Beispiele zur Differentialtopologie von Singularitaten”, Invent. Math., 2:1 (1966), 1–14 ; E. Briskorn, “Primery iz differentsialnoi topologii mnogoobrazii s osobennostyami”, Matematika, 11:6 (1967), 133–144 | DOI | MR | Zbl

[2] Cavicchioli A., Hegenbarth F., Kim A., “On cyclic branched coverings of torus knots”, J. Geometry, 64 (1999), 55–66 | DOI | MR | Zbl

[3] Howie J., Williams G., “Fibonacci type presentations and 3-manifolds”, Topology Appl., 215 (2017), 24–34 | DOI | MR | Zbl

[4] Hempel J., 3-manifolds, Annals Math. Studies, 86, Princeton University Press, Princeton; N.J., 1976, 195 pp. | MR | Zbl

[5] Matveev S., Algorithmic topology and classification of 3-manifolds, Algorithms Comput. Math., 9, 2nd ed., Springer, Berlin, 2007, 492 pp. | DOI | MR | Zbl

[6] Matveev S.V., “Tabulirovanie trekhmernykh mnogoobrazii”, Uspekhi mat. nauk, 60:4 (2005), 97–122 | DOI | MR | Zbl

[7] Three-manifold Recognizer, The computer program developed by the research group of S. Matveev in the department of computer topology and algebra of Chelyabinsk State University

[8] Weber C., Seifert H., “Die Beiden Dodekaederaume”, Math. Z., 37 (1933), 237–253 | DOI | MR

[9] Matveev S.V., Fomenko A.T., “Izoenergeticheskie poverkhnosti gamiltonovykh sistem, perechislenie trekhmernykh mnogoobrazii v poryadke vozrastaniya ikh slozhnosti i vychislenie ob'emov zamknutykh giperbolicheskikh mnogoobrazii”, Uspekhi mat. nauk, 43:1 (1988), 5–22 | MR | Zbl

[10] Weeks J., Hyperbolic structures on 3-manifolds, Thesis (Ph.D.)-Princeton University, Princeton University, Princeton, 1985, 83 pp. | MR

[11] Mednykh A., Vesnin A., “Visualization of the isometry group action on the Fomenko-Matveev-Weeks manifold”, J. Lie Theory, 8:1 (1998), 51–66 | MR | Zbl

[12] Helling H., Kim A., Mennicke J., “A geometric study of Fibonacci groups”, J. Lie Theory, 8:4 (1998), 1–23 | MR | Zbl

[13] Sieradski A.J., “Combinatorial squashings, 3-manifolds, and the third homology of groups”, Invent. Math., 84 (1986), 121–139 | DOI | MR | Zbl

[14] Milnor J., Singular points of complex hypersurfaces, Annals of Mathematics Studies, Princeton University Press and Tokyo University Press, Princeton, 1968, 130 pp. ; Dzh. Milnor, Osobye tochki kompleksnykh giperpoverkhnostei, Mir, M., 1971, 126 pp. | MR | Zbl | MR

[15] Milnor J., “On the 3-dimensional Brieskorn manifolds M(p,q,r)”, Knots, Groups and 3-Manifolds, Ann. of Math. Studies, 84, ed. ed. L. P. Neuwirth, Princeton Univ. Press, Princeton, N. J., 1975, 175–225 | MR

[16] Johnson D., Topics in the theory of group presentations, London Math. Soc. Lect. Note Ser., 42, Cambridge Univ. Press, Cambridge, 1980, 320 pp. | MR | Zbl

[17] Bardakov V.G., Vesnin A.Yu., “Ob obobschenii grupp Fibonachchi”, Algebra i logika, 42:2 (2003), 131–160 | MR | Zbl

[18] Maclachlan C., “Generalizations of Fibonacci numbers, groups and manifolds”, Combinatorial and Geometric Group Theory (Edinburgh, 1993), London Math. Soc. Lect. Note Ser., 204, Cambridge Univ. Press, Cambridge, 1995, 233–238 | MR | Zbl

[19] Johnson D.J., Wamsley J.W., Wright D., “The Fibonacci groups”, Proc. London Math. Soc., s3-29:4 (1974), 577–592 | DOI | MR | Zbl

[20] Szczepanski A., “High dimensional knot groups and HNN extensions of the Fibonacci groups”, J. Knot Theory Ramifications, 7:4 (1998), 503–508 | DOI | MR | Zbl

[21] Campbell C.M., Robertson E.F., “A class of finitely presented groups of Fibonacci type”, J. London Math. Soc., s2-11:2 (1975), 249–255 | DOI | MR | Zbl

[22] Szczepanski A., Vesnin A., “On generalized Fibonacci groups with odd number of generators”, Communications in Algebra, 28:2 (2000), 959–965 | DOI | MR | Zbl

[23] Szczepanski A., Vesnin A., “Generalized Neuwirth Groups and Seifert fibered manifolds”, Algebra Colloquium, 7:3 (2000), 295–303 | DOI | MR | Zbl

[24] Neuwirth L., “An algorithm for the construction of 3-manifolds from 2-complexes”, Proc. Camb. Philos. Soc., 64 (1968), 603–613 | DOI | MR | Zbl

[25] Johnson D.L., Mawdesley H., “Some groups of Fibonacci type”, J. Aust. Math. Soc., 20:2 (1975), 199–204 | DOI | MR | Zbl

[26] Gilbert N., Howie J., “LOG groups and cyclically presented groups”, J. Algebra, 174:1 (1995), 118–131 | DOI | MR | Zbl

[27] Kim A.C., Vesnin A., “Cyclically presented groups and Takahashi manifolds”, Analysis of discrete groups, II (Kyoto, 1996), RIMS Kokyuroku, 1022, 1997, 200–212 | MR | Zbl

[28] Singer J., “Three-dimensional manifolds and their Heegaard diagrams”, Trans. Amer. Math. Soc., 35:1 (1933), 88–111 | DOI | MR

[29] Zeifert G., Trelfall V., Topologiya, Izd-vo ONTI, M., 1938, 400 pp.