The Classification and the Conjugacy Classes of the Finite Subgroups of the Sphere Braid Groups Daciberg Gonçalves, John Guaschi

Total Page:16

File Type:pdf, Size:1020Kb

The Classification and the Conjugacy Classes of the Finite Subgroups of the Sphere Braid Groups Daciberg Gonçalves, John Guaschi The classification and the conjugacy classes of the finite subgroups of the sphere braid groups Daciberg Gonçalves, John Guaschi To cite this version: Daciberg Gonçalves, John Guaschi. The classification and the conjugacy classes of the finite subgroups of the sphere braid groups. Algebraic and Geometric Topology, Mathematical Sciences Publishers, 2008, 8 (2), pp.757-785. 10.2140/agt.2008.8.757. hal-00191422 HAL Id: hal-00191422 https://hal.archives-ouvertes.fr/hal-00191422 Submitted on 26 Nov 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The classification and the conjugacy classes of the finite subgroups of the sphere braid groups DACIBERG LIMA GONC¸ALVES Departamento de Matem´atica - IME-USP, Caixa Postal 66281 - Ag. Cidade de S˜ao Paulo, CEP: 05311-970 - S˜ao Paulo - SP - Brazil. e-mail: [email protected] JOHN GUASCHI Laboratoire de Math´ematiques Nicolas Oresme UMR CNRS 6139, Universit´ede Caen BP 5186, 14032 Caen Cedex, France. e-mail: [email protected] 21st November 2007 Abstract Let n ¥ 3. We classify the finite groups which are realised as subgroups of the sphere braid 2 Õ group Bn ÔS . Such groups must be of cohomological period 2 or 4. Depending on the 2 Ô Õ ¡ Õ value of n, we show that the following are the maximal finite subgroups of Bn S : Z2Ô n 1 ; ¡ Õ the dicyclic groups of order 4n and 4Ô n 2 ; the binary tetrahedral group T1; the binary octahedral group O1; and the binary icosahedral group I. We give geometric as well as 2 Õ some explicit algebraic constructions of these groups in Bn ÔS , and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi’s classification of 2 2 Õ Ô Õ the torsion elements of Bn ÔS , and explain how the finite subgroups of Bn S are related 2 Õ to this classification, as well as to the lower central and derived series of Bn ÔS . 1 Introduction The braid groups Bn of the plane were introduced by E. Artin in 1925 [A1, A2]. Braid groups of surfaces were studied by Zariski [Z]. They were later generalised by Fox to hal-00191422, version 1 - 26 Nov 2007 braid groups of arbitrary topological spaces via the following definition [FoN]. Let M be a compact, connected surface, and let n È N. We denote the set of all ordered n-tuples of distinct points of M, known as the nth configuration space of M, by: ! µ Ô Õ Õ È Fn Ô M p1,..., pn pi M and pi pj if i j . 2000 AMS Subject Classification: 20F36 (primary), 20F50, 20E45, 57M99 (secondary). 1 Configuration spaces play an important rˆole in several branches of mathematics and have been extensively studied, see [CG, FH] for example. Õ The symmetric group Sn on n letters acts freely on Fn Ô M by permuting coordinates. th Õ Ô Õ The corresponding quotient will be denoted by Dn Ô M . The n pure braid group Pn M th Õ Ô Õ (respectively the n braid group Bn Ô M ) is defined to be the fundamental group of Fn M Õ (respectively of Dn Ô M ). Together with the real projective plane RP2, the braid groups of the 2-sphere S2 are of particular interest, notably because they have non-trivial centre [GVB, GG1], and torsion elements [VB, Mu]. Indeed, Van Buskirk showed that among the braid 2 2 Õ Ô Õ groups of compact, connected surfaces, Bn ÔS and Bn RP are the only ones to have 2 Õ torsion [VB]. Let us recall briefly some of the properties of Bn ÔS [FVB, GVB, VB]. 2 2 2 ÝÑ Ô Õ If D S is a topological disc, there is a group homomorphism ι : Bn Bn S Ô Õ induced by the inclusion. If β È Bn, we shall denote its image ι β simply by β. Then 2 Ô Õ Bn S is generated by σ1,..., σn ¡1 which are subject to the following relations: ¡ ¥ ¤ ¤ ¡ σiσj σjσi if |i j| 2and 1 i, j n 1 ¤ ¤ ¡ σiσi 1σi σi 1σiσi 1 for all 1 i n 2, and 2 ¤ ¤ ¤ ¤ ¤ ¤ ¡ ¡ σ1 σn 2σn ¡1σn 2 σ1 1. 2 Õ Consequently, Bn ÔS is a quotient of Bn. The first three sphere braid groups are finite: 2 2 2 Õ Ô Õ Ô Õ B1 ÔS is trivial, B2 S is cyclic of order 2, and B3 S is a ZS-metacyclic group (a group whose Sylow subgroups, commutator subgroup and commutator quotient group are all cyclic) of order 12, isomorphic to the semi-direct product Z3 « Z4 of cyclic groups, the action being the non-trivial one, which in turn is isomorphic to the dicyclic group Dic12 2 Ô Õ ¡ Õ of order 12. The Abelianisation of Bn S is isomorphic to the cyclic group Z2Ô n 1 . The 2 Ô Õ ÝÑ Ô Õ ¡ Õ kernel of the associated projection ξ : Bn S Z2Ô n 1 (which is defined by ξ σi 1 ¡ © 2 2 ¤ ¡ Ô Õ È Ô Õ Ô Õ ¤ Γ for all 1 i n 1) is the commutator subgroup 2 Bn S . If w Bn S then ξ w ¡ Õ is the exponent sum (relative to the σi) of w modulo 2Ô n 1 . 2 È Ô Õ Gillette and Van Buskirk showed that if n ¥ 3 and k N then Bn S has an element ¡ Õ Ô ¡ Õ of order k if and only if k divides one of 2n, 2Ô n 1 or 2 n 2 [GVB]. The torsion 2 2 2 Õ Ô Õ Ô Õ elements of Bn ÔS and Bn RP were later characterised by Murasugi [Mu]. For Bn S , these elements are as follows: 2 Ô Õ THEOREM 1 ([MU]). Let n ¥ 3. Then the torsion elements of Bn S are precisely powers of conjugates of the following three elements: ¤ ¤ ¤ ¡ (a) α0 σ1 σn 2σn ¡1 (which is of order 2n). 2 ¤ ¤ ¤ Ô ¡ Õ (b) α σ σ ¡ σ (of order 2 n 1 ). 1 1 n 2 n ¡1 2 ¤ ¤ ¤ Ô ¡ Õ (c) α σ σ ¡ σ (of order 2 n 2 ). 2 1 n 3 n ¡2 th th th ¡ Õ Ô ¡ Õ The three elements α0, α1 and α2 are respectively n , Ô n 1 and n 2 roots of 2 n Õ Ô ¤ ¤ ¤ Õ ∆ ∆ Ô ∆ n, where n is the so-called ‘full twist’ braid of Bn S , defined by n σ1 σn ¡1 . 2 Ô Õ ¡ Õ Ô ¡ Õ So Bn S admits finite cyclic subgroups isomorphic to Z2n, Z2Ô n 1 and Z2 n 2 . In [GG2], 2 Õ ¥ Ô ∆ we showed that Bn S is generated by α0 and α1. If n 3, n is the unique element 2 2 Õ Ô Õ of Bn ÔS of order 2, and it generates the centre of Bn S . It is also the square of the Garside element (or ‘half twist’) defined by: Ô ¤ ¤ ¤ ÕÔ ¤ ¤ ¤ Õ ¤ ¤ ¤ Ô Õ ¡ Tn σ1 σn ¡1 σ1 σn 2 σ1σ2 σ1. 2 2 Ô Õ For n ¥ 4, Bn S is infinite. It is an interesting question as to which finite groups are 2 Õ Ü Ý realised as subgroups of Bn ÔS (apart of course from the cyclic groups αi and their subgroups given in Theorem 1). Another question is the following: how many conju- 2 Õ gacy classes are there in Bn ÔS of a given abstract finite group? As a partial answer to 2 Õ the first question, we proved in [GG2] that Bn ÔS contains an isomorphic copy of the 2 Õ finite group B3 ÔS of order 12 if and only if n 1 mod 3. While studying the lower central and derived series of the sphere braid groups, we ¡ © 2 Õ Γ Ô showed that 2 B4 S is isomorphic to a semi-direct product of Q8 by a free group of rank 2 [GG3]. After having proved this result, we noticed that the question of the 2 Õ realisation of Q8 as a subgroup of Bn ÔS had been explicitly posed by R. Brown [ATD] in connection with the Dirac string trick [F, N] and the fact that the fundamental group Õ of SOÔ3 is isomorphic to Z2. The case n 4 was studied by J. G. Thompson [ThJ]. In a previous paper, we provided a complete answer to this question: ¥ THEOREM 2 ([GG4]). Let n È N, n 3. 2 Õ (a) Bn ÔS contains a subgroup isomorphic to Q8 if and only if n is even. ¡ © 2 Õ Γ Ô (b) If n is divisible by 4 then 2 Bn S contains a subgroup isomorphic to Q8. As we also pointed out in [GG4], for all n ¥ 3, the construction of Q8 may be gener- 2 Ý Ô Õ alised in order to obtain a subgroup Ü α0, Tn of Bn S isomorphic to the dicyclic group Dic4n of order 4n. 2 Õ It is thus natural to ask which other finite groups are realised as subgroups of Bn ÔS . One common property of the above subgroups is that they are finite periodic groups 2 Õ of cohomological period 2 or 4.
Recommended publications
  • The Classification and the Conjugacy Classesof the Finite Subgroups of The
    Algebraic & Geometric Topology 8 (2008) 757–785 757 The classification and the conjugacy classes of the finite subgroups of the sphere braid groups DACIBERG LGONÇALVES JOHN GUASCHI Let n 3. We classify the finite groups which are realised as subgroups of the sphere 2 braid group Bn.S /. Such groups must be of cohomological period 2 or 4. Depend- ing on the value of n, we show that the following are the maximal finite subgroups of 2 Bn.S /: Z2.n 1/ ; the dicyclic groups of order 4n and 4.n 2/; the binary tetrahedral group T ; the binary octahedral group O ; and the binary icosahedral group I . We give geometric as well as some explicit algebraic constructions of these groups in 2 Bn.S / and determine the number of conjugacy classes of such finite subgroups. We 2 also reprove Murasugi’s classification of the torsion elements of Bn.S / and explain 2 how the finite subgroups of Bn.S / are related to this classification, as well as to the 2 lower central and derived series of Bn.S /. 20F36; 20F50, 20E45, 57M99 1 Introduction The braid groups Bn of the plane were introduced by E Artin in 1925[2;3]. Braid groups of surfaces were studied by Zariski[41]. They were later generalised by Fox to braid groups of arbitrary topological spaces via the following definition[16]. Let M be a compact, connected surface, and let n N . We denote the set of all ordered 2 n–tuples of distinct points of M , known as the n–th configuration space of M , by: Fn.M / .p1;:::; pn/ pi M and pi pj if i j : D f j 2 ¤ ¤ g Configuration spaces play an important roleˆ in several branches of mathematics and have been extensively studied; see Cohen and Gitler[9] and Fadell and Husseini[14], for example.
    [Show full text]
  • INTEGRAL CAYLEY GRAPHS and GROUPS 3 of G on W
    INTEGRAL CAYLEY GRAPHS AND GROUPS AZHVAN AHMADY, JASON P. BELL, AND BOJAN MOHAR Abstract. We solve two open problems regarding the classification of certain classes of Cayley graphs with integer eigenvalues. We first classify all finite groups that have a “non-trivial” Cayley graph with integer eigenvalues, thus solving a problem proposed by Abdollahi and Jazaeri. The notion of Cayley integral groups was introduced by Klotz and Sander. These are groups for which every Cayley graph has only integer eigenvalues. In the second part of the paper, all Cayley integral groups are determined. 1. Introduction A graph X is said to be integral if all eigenvalues of the adjacency matrix of X are integers. This property was first defined by Harary and Schwenk [9] who suggested the problem of classifying integral graphs. This problem ignited a signifi- cant investigation among algebraic graph theorists, trying to construct and classify integral graphs. Although this problem is easy to state, it turns out to be extremely hard. It has been attacked by many mathematicians during the last forty years and it is still wide open. Since the general problem of classifying integral graphs seems too difficult, graph theorists started to investigate special classes of graphs, including trees, graphs of bounded degree, regular graphs and Cayley graphs. What proves so interesting about this problem is that no one can yet identify what the integral trees are or which 5-regular graphs are integral. The notion of CIS groups, that is, groups admitting no integral Cayley graphs besides complete multipartite graphs, was introduced by Abdollahi and Jazaeri [1], who classified all abelian CIS groups.
    [Show full text]
  • Generalized Quaternions
    GENERALIZED QUATERNIONS KEITH CONRAD 1. introduction The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomor- phism). The other one, D4, can be constructed as a semi-direct product: ∼ ∼ × ∼ D4 = Aff(Z=(4)) = Z=(4) o (Z=(4)) = Z=(4) o Z=(2); where the elements of Z=(2) act on Z=(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We will see how this is done in Section2 and then jazz up the construction in Section3 to make an infinite family of similar groups with Q8 as the simplest member. In Section4 we will compare this family with the dihedral groups and see how it fits into a bigger picture. 2. The quaternion group from a semi-direct product The group Q8 is built out of its subgroups hii and hji with the overlapping condition i2 = j2 = −1 and the conjugacy relation jij−1 = −i = i−1. More generally, for odd a we have jaij−a = −i = i−1, while for even a we have jaij−a = i. We can combine these into the single formula a (2.1) jaij−a = i(−1) for all a 2 Z. These relations suggest the following way to construct the group Q8. Theorem 2.1. Let H = Z=(4) o Z=(4), where (a; b)(c; d) = (a + (−1)bc; b + d); ∼ The element (2; 2) in H has order 2, lies in the center, and H=h(2; 2)i = Q8.
    [Show full text]
  • A Symplectic Resolution for the Binary Tetrahedral Group
    A SYMPLECTIC RESOLUTION FOR THE BINARY TETRAHEDRAL GROUP MANFRED LEHN & CHRISTOPH SORGER Abstract. We describe an explicit symplectic resolution for the quo- tient singularity arising from the four-dimensional symplectic represen- ation of the binary tetrahedral group. Let G be a nite group with a complex symplectic representation V . The symplectic form σ on V descends to a symplectic form σ¯ on the open regular part of V=G. A proper morphism f : Y ! V=G is a symplectic resolution if Y is smooth and if f ∗σ¯ extends to a symplectic form on Y . It turns out that symplectic resolutions of quotient singularities are a rare phenomenon. By a theorem of Verbitsky [9], a necessary condition for the existence of a symplectic resolution is that G be generated by symplectic reections, i.e. by elements whose x locus on V is a linear subspace of codimension 2. Given an arbitrary complex representation V0 of a nite group G, we obtain a symplectic representation on ∗, where ∗ denotes the contragradient V0 ⊕V0 V0 representation of V0. In this case, Verbitsky's theorem specialises to an earlier theorem of Kaledin [7]: For ∗ to admit a symplectic resolution, the V0 ⊕ V0 =G action of G on V0 should be generated by complex reections, in other words, V0=G should be smooth. The complex reection groups have been classied by Shephard and Todd [8], the symplectic reection groups by Cohen [2]. The list of Shephard and Todd contains as a sublist the nite Coxeter groups. The question which of these groups G ⊂ Sp(V ) admits a symplectic res- olution for V=G has been solved for the Coxeter groups by Ginzburg and Kaledin [3] and for arbitrary complex reection groups most recently by Bellamy [1].
    [Show full text]
  • Maximal Subgroups of the Coxeter Group W(H4) and Quaternions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 412 (2006) 441–452 www.elsevier.com/locate/laa Maximal subgroups of the Coxeter group W(H4) and quaternions Mehmet Koca a, Ramazan Koç b,∗, Muataz Al-Barwani a, Shadia Al-Farsi a aDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Muscat, Oman bDepartment of Physics, Faculty of Engineering, Gaziantep University, 27310 Gaziantep, Turkey Received 9 October 2004; accepted 17 July 2005 Available online 12 September 2005 Submitted by H. Schneider Abstract The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H4) of order 14,400. Its derived subgroup is the largest finite subgroup W(H4)/Z2 of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W(H2) × W(H2)]Z4 and W(H3) × Z2 possess non- crystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H4) with sets of quaternion pairs acting on the quaternionic root systems. © 2005 Elsevier Inc. All rights reserved. AMS classification: 20G20; 20F05; 20E34 Keywords: Structure of groups; Quaternions; Coxeter groups; Subgroup structure ∗ Corresponding author. Tel.: +90 342 360 1200; fax: +90 342 360 1013.
    [Show full text]
  • 12.6 Further Topics on Simple Groups 387 12.6 Further Topics on Simple Groups
    12.6 Further Topics on Simple groups 387 12.6 Further Topics on Simple Groups This Web Section has three parts (a), (b) and (c). Part (a) gives a brief descriptions of the 56 (isomorphism classes of) simple groups of order less than 106, part (b) provides a second proof of the simplicity of the linear groups Ln(q), and part (c) discusses an ingenious method for constructing a version of the Steiner system S(5, 6, 12) from which several versions of S(4, 5, 11), the system for M11, can be computed. 12.6(a) Simple Groups of Order less than 106 The table below and the notes on the following five pages lists the basic facts concerning the non-Abelian simple groups of order less than 106. Further details are given in the Atlas (1985), note that some of the most interesting and important groups, for example the Mathieu group M24, have orders in excess of 108 and in many cases considerably more. Simple Order Prime Schur Outer Min Simple Order Prime Schur Outer Min group factor multi. auto. simple or group factor multi. auto. simple or count group group N-group count group group N-group ? A5 60 4 C2 C2 m-s L2(73) 194472 7 C2 C2 m-s ? 2 A6 360 6 C6 C2 N-g L2(79) 246480 8 C2 C2 N-g A7 2520 7 C6 C2 N-g L2(64) 262080 11 hei C6 N-g ? A8 20160 10 C2 C2 - L2(81) 265680 10 C2 C2 × C4 N-g A9 181440 12 C2 C2 - L2(83) 285852 6 C2 C2 m-s ? L2(4) 60 4 C2 C2 m-s L2(89) 352440 8 C2 C2 N-g ? L2(5) 60 4 C2 C2 m-s L2(97) 456288 9 C2 C2 m-s ? L2(7) 168 5 C2 C2 m-s L2(101) 515100 7 C2 C2 N-g ? 2 L2(9) 360 6 C6 C2 N-g L2(103) 546312 7 C2 C2 m-s L2(8) 504 6 C2 C3 m-s
    [Show full text]
  • Platonic Solids Generate Their Four-Dimensional Analogues.', Acta Crystallographica Section A., 69 (6)
    Durham Research Online Deposited in DRO: 26 January 2014 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Dechant, Pierre-Philippe (2013) 'Platonic solids generate their four-dimensional analogues.', Acta crystallographica section A., 69 (6). pp. 592-602. Further information on publisher's website: http://dx.doi.org/10.1107/S0108767313021442 Publisher's copyright statement: Additional information: Published on behalf of the International Union of Crystallography. Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk electronic reprint Acta Crystallographica Section A Foundations of Crystallography ISSN 0108-7673 Platonic solids generate their four-dimensional analogues Pierre-Philippe Dechant Acta Cryst. (2013). A69, 592–602 Copyright c International Union of Crystallography Author(s) of this paper may load this reprint on their own web site or institutional repository provided that this cover page is retained. Republication of this article or its storage in electronic databases other than as specified above is not permitted without prior permission in writing from the IUCr.
    [Show full text]
  • Platonic Solids Generate Their Four-Dimensional Analogues
    This is a repository copy of Platonic solids generate their four-dimensional analogues. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/85590/ Version: Accepted Version Article: Dechant, Pierre-Philippe orcid.org/0000-0002-4694-4010 (2013) Platonic solids generate their four-dimensional analogues. Acta Crystallographica Section A : Foundations of Crystallography. pp. 592-602. ISSN 1600-5724 https://doi.org/10.1107/S0108767313021442 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ 1 Platonic solids generate their four-dimensional analogues PIERRE-PHILIPPE DECHANT a,b,c* aInstitute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, United Kingdom, bPhysics Department, Arizona State University, Tempe, AZ 85287-1604, United States, and cMathematics Department, University of York, Heslington, York, YO10 5GG, United Kingdom. E-mail: [email protected] Polytopes; Platonic Solids; 4-dimensional geometry; Clifford algebras; Spinors; Coxeter groups; Root systems; Quaternions; Representations; Symmetries; Trinities; McKay correspondence Abstract In this paper, we show how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone.
    [Show full text]
  • Remarks on the Cohomology of Finite Fundamental Groups of 3–Manifolds
    Geometry & Topology Monographs 14 (2008) 519–556 519 arXiv version: fonts, pagination and layout may vary from GTM published version Remarks on the cohomology of finite fundamental groups of 3–manifolds SATOSHI TOMODA PETER ZVENGROWSKI Computations based on explicit 4–periodic resolutions are given for the cohomology of the finite groups G known to act freely on S3 , as well as the cohomology rings of the associated 3–manifolds (spherical space forms) M = S3=G. Chain approximations to the diagonal are constructed, and explicit contracting homotopies also constructed for the cases G is a generalized quaternion group, the binary tetrahedral group, or the binary octahedral group. Some applications are briefly discussed. 57M05, 57M60; 20J06 1 Introduction The structure of the cohomology rings of 3–manifolds is an area to which Heiner Zieschang devoted much work and energy, especially from 1993 onwards. This could be considered as part of a larger area of his interest, the degrees of maps between oriented 3– manifolds, especially the existence of degree one maps, which in turn have applications in unexpected areas such as relativity theory (cf Shastri, Williams and Zvengrowski [41] and Shastri and Zvengrowski [42]). References [1,6,7, 18, 19, 20, 21, 22, 23] in this paper, all involving work of Zieschang, his students Aaslepp, Drawe, Sczesny, and various colleagues, attest to his enthusiasm for these topics and the remarkable energy he expended studying them. Much of this work involved Seifert manifolds, in particular, references [1, 6, 7, 18, 20, 23]. Of these, [6, 7, 23] (together with [8, 9]) successfully completed the programme of computing the ring structure H∗(M) for any orientable Seifert manifold M with 1 2 3 3 G := π1(M) infinite.
    [Show full text]
  • The Fundamental Group of SO(N) Via Quotients of Braid Groups Arxiv
    The Fundamental Group of SO(n) Via Quotients of Braid Groups Ina Hajdini∗ and Orlin Stoytchevy July 21, 2016 Abstract ∼ We describe an algebraic proof of the well-known topological fact that π1(SO(n)) = Z=2Z. The fundamental group of SO(n) appears in our approach as the center of a certain finite group defined by generators and relations. The latter is a factor group of the braid group Bn, obtained by imposing one additional relation and turns out to be a nontrivial central extension by Z=2Z of the corresponding group of rotational symmetries of the hyperoctahedron in dimension n. 1 Introduction. n The set of all rotations in R forms a group denoted by SO(n). We may think of it as the group of n × n orthogonal matrices with unit determinant. As a topological space it has the structure of a n2 smooth (n(n − 1)=2)-dimensional submanifold of R . The group structure is compatible with the smooth one in the sense that the group operations are smooth maps, so it is a Lie group. The space SO(n) when n ≥ 3 has a fascinating topological property—there exist closed paths in it (starting and ending at the identity) that cannot be continuously deformed to the trivial (constant) path, but going twice along such a path gives another path, which is deformable to the trivial one. For example, if 3 you rotate an object in R by 2π along some axis, you get a motion that is not deformable to the trivial motion (i.e., no motion at all), but a rotation by 4π is deformable to the trivial motion.
    [Show full text]
  • Half-Bps M2-Brane Orbifolds 3
    HALF-BPS M2-BRANE ORBIFOLDS PAUL DE MEDEIROS AND JOSÉ FIGUEROA-O’FARRILL Abstract. Smooth Freund–Rubin backgrounds of eleven-dimensional supergravity of the form AdS X7 and preserving at least half of the supersymmetry have been recently clas- 4 × sified. Requiring that amount of supersymmetry forces X to be a spherical space form, whence isometric to the quotient of the round 7-sphere by a freely-acting finite subgroup of SO(8). The classification is given in terms of ADE subgroups of the quaternions embed- dedin SO(8) as the graph of an automorphism. In this paper we extend this classification by dropping the requirement that the background be smooth, so that X is now allowed to be an orbifold of the round 7-sphere. We find that if the background preserves more than half of the supersymmetry, then it is automatically smooth in accordance with the homo- geneity conjecture, but that there are many half-BPS orbifolds, most of them new. The classification is now given in terms of pairs of ADE subgroups of quaternions fibred over the same finite group. We classify such subgroups and then describe the resulting orbi- folds in terms of iterated quotients. In most cases the resulting orbifold can be described as a sequence of cyclic quotients. Contents List of Tables 2 1. Introduction 3 How to use this paper 5 2. Spherical orbifolds 7 2.1. Spin orbifolds 8 2.2. Statement of the problem 9 3. Finite subgroups of the quaternions 10 4. Goursat’s Lemma 12 arXiv:1007.4761v3 [hep-th] 25 Aug 2010 4.1.
    [Show full text]
  • Convex Polytopes and Tilings with Few Flag Orbits
    Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.
    [Show full text]