DOCSLIB.ORG

On Metacyclic Extensions Tome 24, No 2 (2012), P

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Metacyclic Extensions Tome 24, No 2 (2012), P Masanari KIDA On metacyclic extensions Tome 24, no 2 (2012), p. 339-353. <http://jtnb.cedram.org/item?id=JTNB_2012__24_2_339_0> © Société Arithmétique de Bordeaux, 2012, tous droits réservés. L’accès aux articles de la revue « Journal de Théorie des Nom- bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb.cedram. org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Journal de Théorie des Nombres de Bordeaux 24 (2012), 339-353 On metacyclic extensions par Masanari KIDA Résumé. Nous construisons des extensions galoisiennes de groupes de Galois métacycliques variés au moyen d’une théorie de Kummer émanant d’une isogénie de certains tores algébriques. En particulier, notre méthode nous permet de construire des tores algébriques paramétrant des extensions métacycliques. Abstract. Galois extensions with various metacyclic Galois groups are constructed by means of a Kummer theory arising from an isogeny of certain algebraic tori. In particular, our method enables us to construct algebraic tori parameterizing metacyclic extensions. 1. Introduction A finite group G is called metacyclic if it contains a normal cyclic sub- group N such that the quotient group G/N is also cyclic. The category of metacyclic groups contains important families of groups such as dihedral groups and Frobenius groups. A Galois extension L/k is called a metacyclic extension if the Galois group Gal(L/k) is isomorphic to a metacyclic group. In this paper, we give a method to construct metacyclic extensions using a Kummer theory without roots of unity studied in our previous paper [10]. Évariste Galois already knew that if a polynomial of prime degree `(≥ 5) is solvable, then the Galois group of the polynomial is a Frobenius group (see [7, Lemma 7.1.2]). Thus our method enables us to construct such solvable extensions. In their paper [14], Nakano and Sase also study a construction of metacyclic extensions, which is a generalization of a former result of Imaoka and Kishi [6] on Frobenius and dihedral extensions. Our construction supersedes their construction in some respect. Furthermore, because our construction comes from a Kummer theory and there are geo- metric objects parameterizing the field extensions, we can expect to get more algebraic and arithmetic information about the extensions. After we give some preliminaries on metacyclic groups and the Kummer theory in the next section, we state and prove one of our main theorem in Section3. In Section4, we explain geometry behind our construction. In Manuscrit reçu le 26 janvier 2011. This research is supported in part by Grant-in-Aid for Scientific Research (C) (No. 19540015), Ministry of Education, Science, Sports and Culture, Japan. 340 Masanari Kida fact, we show that our metacyclic extensions are parameterized by rational points on certain algebraic tori defined over the base field. In Section5, we give a relationship between our construction and the construction due to Sase and Nakano. Several examples are given in the final section. Notation. The following symbols are used throughout this paper. Let ` be an odd prime number fixed once for all. We denote a field of ` elements by F`. We also fix a base field k whose characteristic is different from `. We assume that any separable extensions of k are contained in a fixed separable closure ksep of k. For an integer m prime to the characteristic of k, we denote by ζm a primitive m-th root of unity in ksep. For any separable extension K over k, we write Kc for the `-th cyclotomic extension K(ζ`). Let r be a divisor of ` − 1 and F a cyclic extension over k of degree r r = [F : k]. We denote by σ a generator of the Galois group of F/k: Gal(F/k) = hσi. Let n be the degree of the cyclotomic extension Fc/F : n = [Fc : F ]; and τ a generator of the Galois group of Fc/F : Gal(Fc/F ) = hτi. Since both kc/k and F/k are abelian extensions, Fc = F kc is also an abelian extension over k of degree nr. Therefore there exists a subgroup of Gal(Fc/k) isomorphic to Gal(F/k) whose generator is a lift of σ. We denote the generator also by σ. Then we have an isomorphism ∼ Gal(Fc/k) = hσi × hτi. For a finite group G, a field extension L/k is called a G-extension if it is a Galois extension whose Galois group is isomorphic to G. Other notation will be introduced when we need it. 2. Preliminaries In this section, we give some preliminaries from group theory and a Kummer theory for certain algebraic tori. 2.1. Metacyclic groups. We refer to [3, Section 47] for basic facts on metacyclic groups. A metacyclic group G is an extension of a cyclic group with cyclic kernel N: 1 −−−−→ N −−−−→ G −−−−→ G/N −−−−→ 1 (exact) In this paper, we are solely concerned with the case where the order of N is a prime number `. We write N = hρi and G/N = hσNi. Since N is normal On metacyclic extensions 341 × −1 x in G, we can find an element x in F` such that σ ρσ = ρ holds. For a −m m xm × positive integer m, we have σ ρσ = ρ . Let s be the order of x in F` and r the order of σN. Then it follows that (2.1) s | r | ` − 1. Hence the extension splits by a theorem of Schur [4, Theorem 15.2.2] and there exists a subgroup H of G isomorphic to G/N satisfying G = N o 0 × H. Note that if x and x generate the same subgroup in F` , then the corresponding metacyclic groups are isomorphic. Therefore such a group is determined by `, r and s and thus we denote this group by M`(r, s): r ` −1 x M`(r, s) = hσ, ρ | σ = 1, ρ = 1, σ ρσ = ρ , ord(x mod `) = si. It is easy to observe that M`(r, s) is abelian ⇐⇒ s = 1; M`(r, s) is a Frobenius group F`r ⇐⇒ s = r; M`(r, s) is a dihedral group D` of order 2` ⇐⇒ s = r = 2. 2.2. Kummer theory via algebraic tori. Suppose that there exists an integer-coefficient polynomial n−1 (2.2) P(t) = c1 + c2t + ··· + cnt ∈ Z[t] of degree n − 1 satisfying the following two conditions: ∼ (2.3) Z[ζn]/(P(ζn)) = Z/(`) = F`; and i i × (2.4) P(ζn) ∈ Z[ζn] for all i with (n, i) > 1. We assume that the ring isomorphism (2.3) induces a group isomorphism ∼ (2.5) νk : Gal(kc/k) → hζn mod P(ζn)i. Therefore we have n = [kc : k]. Let Rkc/kGm be the Weil restriction of scalars of the multiplicative group. The algebraic torus Rkc/kGm is an n- dimensional torus defined over k splitting over kc. For other basic properties of Rkc/kGm, see [15, 3.12]. Then the circulant matrix c1 c2 ··· cn cn c1 ··· cn−1 (2.6) circ(c1, c2, . , cn) = . . ··· . c2 c3 ··· c1 defines an endomorphism Λ of degree ` on the character module RF\c/kGm of Rkc/kGm, which is, by definition, RF\c/kGm = Homk-schemes(Rkc/kGm, Gm,k). 342 Masanari Kida In fact, we can show that | det(circ(c1, c2, . , cn))| = ` under our assump- tions. In the dual category, we have a self-isogeny λ of degree ` on the k-torus Rkc/kGm. The following theorem is proved in [10]. Theorem 2.1. Let λ be the self-isogeny of Rkc/kGm of degree ` defined as above. Then every point in the kernel of λ is k-rational and the exact sequence attached to the isogeny λ λ 1 −−−−→ ker λ −−−−→ Rkc/kGm −−−−→ Rkc/kGm −−−−→ 1 (exact) induces the Kummer duality ∼ (2.7) κk : Rkc/kGm(k)/λRkc/kGm(k) → Homcont(Gal(ksep/k), ker λ(k)). For any P ∈ Rkc/kGm(k), the image κk(P ) is a homomorphism χP send- 1−s ing every s ∈ Gal(ksep/k) to P . As a consequence, every cyclic exten- sion over k of degree ` is obtained by adjoining an inverse image λ−1(P ) −1 of some P ∈ Rkc/kGm(k) and a generator ρ of Gal(k(λ (P ))/k) acts by ρ(λ−1(P )) = Zλ−1(P ) with some Z ∈ ker λ(k). It is known that this Kummer theory holds for the following cases: • If n is a prime and if there exists an element λ ∈ Z[ζn] whose norm is `. Then we can find P(t) satisfying our assumptions. • We can always construct P(t) in the case where n = 4. • The base field k of the Kummer theory can be descended to the field Q of rational numbers when ` = 3, 5, 7, and 11. See [10, Section 4] for the detail. 3. Algebraic theorem A field extension L/k is called metacyclic extension if it is a Galois extension whose Galois group is isomorphic to a metacyclic group. Our aim is constructing all metacyclic extensions with Galois group M`(r, s) over k containing a given intermediate field F under certain assumptions. We sometimes refer to such an extension by L/F/k. Here we understand that F is the fixed field by the normal subgroup of order ` in M`(r, s).
Load more

Recommended publications
  • The Conjugation Degree on a Set of Metacyclic 3-Groups and 5-Groups with Their Related Graphs
    THE CONJUGATION DEGREE ON A SET OF METACYCLIC 3-GROUPS AND 5-GROUPS WITH THEIR RELATED GRAPHS SITI NORZIAHIDAYU AMZEE ZAMRI UNIVERSITI TEKNOLOGI MALAYSIA THE CONJUGATION DEGREE ON A SET OF METACYCLIC 3-GROUPS AND 5-GROUPS WITH THEIR RELATED GRAPHS SITI NORZIAHIDAYU AMZEE ZAMRI A thesis submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy Faculty of Science Universiti Teknologi Malaysia SEPTEMBER 2018 iii This thesis is specially dedicated to my dearest family, supervisor, colleagues and friends. iv ACKNOWLEDGEMENT It has been a very tough, memorable yet wonderful experience I had as I was writing and completing my PhD thesis for these three and a half years. For that, I would like to express my deepest gratitude towards my one and only supervisor, Prof Dr Nor Haniza Sarmin, for her unlimited support, patience, guidance and abundance of love as my mother in UTM. Besides, I would also like to thank my co-authors, Dr Sanaa Mohamed Saleh Omer, Dr Mustafa Anis El-Sanfaz and Dr Sanhan Muhammad Salih Khasraw for their help and guidance along my PhD journey. Also, I would like to express my love and thank to my family and UTM friends for their help, care and support. Not to forget my dearest besties, Noraihan Afiqah Rawi and Adnin Afifi Nawi who always had my back and for their never ending love and support. Finally, I would like to thank the Ministry of Higher Education Malaysia for MyPhD scholarship and also to Faculty of Science and Universiti Teknologi Malaysia for the conducive study environment.
    [Show full text]
  • GENERATORS of SPLIT EXTENSIONS of ABELIAN GROUPS by CYCLIC GROUPS3 of Unimodular Rows in M with Size N − 1 Under the Action Of
    GENERATORS OF SPLIT EXTENSIONS OF ABELIAN GROUPS BY CYCLIC GROUPS LUC GUYOT Abstract. Let G ≃ M ⋊ C be an n-generator group which is a split ex- tension of an Abelian group M by a cyclic group C. We study the Nielsen equivalence classes and T-systems of generating n-tuples of G. The sub- group M can be turned into a finitely generated faithful module over a suitable quotient R of the integral group ring of C. When C is infinite, we show that the Nielsen equivalence classes of the generating n-tuples of G correspond bijectively to the orbits of unimodular rows in M n−1 under the action of a subgroup of GLn−1(R). Making no assumption on the cardi- nality of C, we exhibit a complete invariant of Nielsen equivalence in the case M ≃ R. As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag-Solitar groups, split metacyclic groups and lamplighter groups. 1. Introduction 1.1. Nielsen equivalence related to equivalence of unimodular rows. Given a finitely generated group G, we denote by rk(G) the minimal number of its generators. For n ≥ rk(G), we let Vn(G) be the set of generating n- vectors of G, i.e., the set of elements in Gn whose components generate G. In order to classify generating vectors, we can rely on a well-studied equivalence relation on Vn(G), namely the Nielsen equivalence relation: two generating n-vectors are said to be Nielsen equivalent if they can be related by a finite sequence of transformations taken in the set {Lij,Ii;1 ≤ i =6 j ≤ n} where g arXiv:1604.08896v6 [math.GR] 22 Jun 2018 Lij and Ii replace the component gi of = (g1,...,gn) ∈ Vn(G) by gjgi and −1 gi respectively and leave the other components unchanged.
    [Show full text]
  • The Schur Multiplicator of Metacyclic Groups'
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 40, Number 2, October 1973 THE SCHUR MULTIPLICATOR OF METACYCLIC GROUPS' F. RUDOLF BEYL Abstract. The Schur multiplicator Hfi of a (finite) meta- cyclic group G is computed with the help of the Lyndon spectral sequence. The order of Hfi is a useful invariant of G. A metacyclic group with vanishing Schur multiplicator can be presented with two generators and two relators. A simple description of the totality of these groups is given and all such /»-groups are classified. 1. Introduction. All groups in this note are finite. The group G is called metacyclic if it has a cyclic normal subgroup which yields a cyclic factor group. If M and N are natural numbers, and r and X integers, and r'v=l mod M, then we introduce G(M, N, r, X) = (a, b:aM = 1, b • a ■ZT1 = a\ bN = ß*^^'-i>). Obviously, G(M, N, r, X) is supported by the metacyclic extension (1) e(M, N, r, X):TZMàJL^. g(M, N, r, X) -^-» Cv, where CN is the cyclic group of order N and 2,M the additive cyclic group of order M, and the distinguished generator c of C^ operates on Zw by multiplication by r. (We indicate this by the superscript.) Moreover, 77 maps b to c and a to 1 and k embeds Zjf as the cyclic subgroup of G generated by a. It is well known [6, III §7] that all metacyclic groups can be presented as in (1). Our main result is that the Schur multiplicator H2G(M, N, r, X) is the cyclic group of order (X, h(M, N, r)) where (2) h(M, N, r) = M"1 • (M, r - 1) • (M, 1 + r + • ■■ + r^1) Received by the editors January 29, 1973.
    [Show full text]
  • A NOTE on FINITE GROUPS with FEW TI-SUBGROUPS Jiangtao Shi
    International Electronic Journal of Algebra Volume 23 (2018) 42-46 DOI: 10.24330/ieja.373640 A NOTE ON FINITE GROUPS WITH FEW TI-SUBGROUPS Jiangtao Shi, Jingjing Huang and Cui Zhang Received: 10 June 2016; Revised: 29 July 2017; Accepted: 10 August 2017 Communicated by Abdullah Harmancı Abstract. In [Comm. Algebra, 43(2015), 2680-2689], finite groups all of whose metacyclic subgroups are TI-subgroups have been classified by S. Li, Z. Shen and N. Du. In this note we investigate a finite group all of whose non-metacyclic subgroups are TI-subgroups. We prove that G is a group all of whose non-metacyclic subgroups are TI-subgroups if and only if all non- metacyclic subgroups of G are normal. Furthermore, we show that a group all of whose non-cyclic subgroups are TI-subgroups has a Sylow tower. Mathematics Subject Classification (2010): 20D10 Keywords: Non-metacyclic subgroup, TI-subgroup, normal, solvable, Sylow tower 1. Introduction In this paper all groups are assumed to be finite. Let G be a group. If G has a cyclic normal subgroup H such that G=H is cyclic, then G is said to be a metacyclic group. It is obvious that a cyclic group is also a metacyclic group but a metacyclic group might not be a cyclic group. Let G be a group and K a subgroup of G. If Kg \K = 1 or K for all g 2 G, then K is called a TI-subgroup of G. It is clear that every normal subgroup of a group G is a TI-subgroup of G but a TI-subgroup of G might not be a normal subgroup of G.
    [Show full text]
  • Coherence, Subgroup Separability, and Metacyclic Structures for a Class of Cyclically Presented Groups
    Accepted Manuscript Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups William A. Bogley, Gerald Williams PII: S0021-8693(17)30088-1 DOI: http://dx.doi.org/10.1016/j.jalgebra.2017.02.002 Reference: YJABR 16106 To appear in: Journal of Algebra Received date: 29 June 2016 Please cite this article in press as: W.A. Bogley, G. Williams, Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups, J. Algebra (2017), http://dx.doi.org/10.1016/j.jalgebra.2017.02.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups William A.Bogleya, Gerald Williamsb aDepartment of Mathematics, Kidder Hall 368, Oregon State University, Corvallis, OR 97331-4605, USA. bDepartment of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, UK. Abstract We study a class M of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defin- ing relations. This class includes many finite metacyclic generalized Fibonacci groups that have been previously identified in the literature.
    [Show full text]
  • UNITS in Fqk (Cp N. Makhijani, RK Sharma and JB
    International Electronic Journal of Algebra Volume 18 (2015) 21-33 UNITS IN Fqk (Cp or Cq) N. Makhijani, R. K. Sharma and J. B. Srivastava Received: 5 February 2014 Communicated by A. C¸i˘gdem Ozcan¨ k Abstract. Let q be a prime, Fqk be a finite field having q elements and n q −1 r Cn or Cq be a group with presentation h a; b j a ; b ; b ab = a i, where (n; rq) = 1 and q is the multiplicative order of r modulo n. In this paper, we address the problem of computing the Wedderburn decomposition of the group algebra Fqk (Cn or Cq) modulo its Jacobson radical. As a consequence, the structure of the unit group of Fqk (Cp or Cq) is obtained when p is a prime different from q. Mathematics Subject Classification (2010): 16U60, 16S34, 20C05 Keywords: Unit group, group algebra, Wedderburn decomposition 1. Introduction Let FG be the group algebra of a finite group G over a field F and U(FG) be its unit group. The study of the group of units is one of the classical topics in group ring theory and has applications in coding theory (cf. [17, 28]) and cryptography (cf. [18]). Results obtained in this direction are useful for the investigation of Lie properties of group rings, isomorphism problem and other open questions in this area [6]. In [3], Bovdi gave a comprehensive survey of results concerning the group of units of a modular group algebra of characteristic p. There is a long tradition on the study of the unit group of finite group algebras [4, 5, 12{16, 22, 25, 26].
    [Show full text]
  • 10 Iranian Group Theory Conference (IGTC 2018)
    Iranian Group Theory Society Kharazmi University 10th Iranian Group Theory Conference th 10 Iranian Group Theory Conference Extended Abstract Booklet University of Kharazmi (IGTC 2018) Tehran, Iran 24-26 January, 2018 http://igtc.khu.ac.ir/ Iranian Group Theory Society Kharazmi University 10th Iranian Group Theory Conference Scientific Committee Chair: Reza Orfi (Kharazmi University) Alireza Abdollahi (Isfahan University ) Hassan Alavi (Buali Sina University) Alireza Ashrafi (Kashan University) Mohammad Reza Darafsheh (Tehran University ) Hossein Doostie (Kharazmi University) Shirin Fouladi (Kharazmi University) Alireza Jamali (Kharazmi University) Behrouz Khosravi (Amirkabir University of Technology) Alireza Moghaddamfar (K.N.Toosi University of Technology) Hamid Mousavi (Tabriz University ) Mohammad Reza Rajabzadeh Moghaddam (Ferdowsi University ) Mojtaba Ramezan Nassab (Kharazmi Universty) Mohammad Reza Salarian (Kharazmi University) Mohammad Shahriary ( Tabriz University ) Iranian Group Theory Society Kharazmi University 10th Iranian Group Theory Conference Executive Committee Chair: Esmail Babolian Keivan Borna Morteza Essmaili Shahnam Javadi Reza Orfi Mojtaba Ramezan Nassab Amir H. Sanatpour Iranian Group Theory Society Kharazmi University 10th Iranian Group Theory Conference Contents Plenary Talks The classification of p-groups Bettina Eick ……………………………………………………………………… 1 Artin’s conjecture Sophie Morel …………………………………………………………………… 2 A history of the order of the multipliers of p-groups and Lie algebra Mohsen Parvizi ………………………………………………………………….
    [Show full text]
  • Character Tables of Metacyclic Groups
    Glasgow Math. J. 57 (2015) 387–400. C Glasgow Mathematical Journal Trust 2014. doi:10.1017/S0017089514000366. CHARACTER TABLES OF METACYCLIC GROUPS STEPHEN P. HUMPHRIES AND DANE C. SKABELUND Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A. e-mail: [email protected], [email protected] (Received 4 June 2013; accepted 8 April 2014; first published online 17 December 2014) Abstract. We show that two metacyclic groups of the following types are isomorphic if they have the same character tables: (i) split metacyclic groups, (ii) the metacyclic p-groups and (iii) the metacyclic {p, q}-groups, where p, q are odd primes. 2010 Mathematics Subject Classification. Primary 20C15; Secondary 20D15, 20F16, 20F22. 1. Introduction. A group G is metacyclic if it has a cyclic normal subgroup K such that G/K is cyclic. Equivalently, the group G has cyclic subgroups S and K, with K normal in G, such that G = SK. If we can find such H, K with H ∩ K ={1}, then we say that G is split metacyclic.ItwasshownbyHolder¨ [11, Theorem 7.21] that a group is metacyclic if and only if it has a presentation of the form α δ β a γ G = Gα,β,γ,δ =a, b | a = b , b = 1, b = b , where gcd(β,γ) = 1,β | (γ α − 1) and β | δ(γ − 1). However, in general the parameters α, β, γ ,andδ are not invariants of the group G. Note that if δ = 0, then G is split metacyclic. One says that two groups G, G have the same character tables if there is a bijection χ ↔ χ of their irreducible characters and a bijection (g)G ↔ (g)G of their classes such that χ(g) = χ (g).
    [Show full text]
  • The Symmetric Genus of Metacyclic Groups
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector TOPOLOGY AND ITS APPLICATIONS ELSEIVIER Topology and its Applications 66 (1995) 101-I 15 The symmetric genus of metacyclic groups Coy L. May *,I, Jay Zimmerman 2 Department of Mathematics, Towson State University, Baltimore, MD 21204, USA Received 19 July 1993; revised 24 February 1995 Abstract Let G be a finite group. The symmetric genus of G is the minimum genus of any Riemann surface on which G acts faithfully. Here we determine a useful lower bound for the symmetric genus of a finite group with a cyclic quotient group. The lower bound is attained for the family of K-metacyclic groups, and we determine the symmetric genus of each nonabelian subgroup of a K-metacyclic group. We also provide some examples of other groups for which the iower bound is attained. We use the standard representation of a finite group as a quotient of a noneuclidean crystallographic (NEC) group by a Fuchsian surface group. Keywords: Symmetric genus; Riemann surface; Metacyclic group; Noneuclidean crystallographic group; Automorphism group AMS ~Zusszjkation: Primary 57M60, Secondary 20HlO; 3OF3.5 1. Introduction The symmetric genus a(G) [I91 of a finite group G is the minimum genus of any Rieniann surface on which G acts faithfully (possibly reversing orientation). There are now many results on the symmetric genus of various classes of groups. For example, the symmetric genus of a finite abelian group has been completely determined [ 131. The groups which have symmetric genus zero are also well known, and the groups of symmetric genus one have been classified in a sense [6, Section 6.3.31; also see [16].
    [Show full text]
  • Seifert Fiberings
    Mathematical Surveys and Monographs Volume 166 Seifert Fiberings Kyung Bai Lee Frank Raymond American Mathematical Society surv-166-lee3-cov.indd 1 10/21/10 11:26 AM http://dx.doi.org/10.1090/surv/166 Seifert Fiberings Seifert Fiberings Kyung Bai Lee Frank Raymond Mathematical Surveys and Monographs Volume 166 Seifert Fiberings Kyung Bai Lee Frank Raymond American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2000 Mathematics Subject Classification. Primary 55R55, 57S30, 57–XX; Secondary 53C30, 55R91, 58E40, 58D19, 57N16. For additional information and updates on this book, visit www.ams.org/bookpages/surv-166 Library of Congress Cataloging-in-Publication Data Lee, Kyung Bai, 1943– Seifert fiberings / Kyung Bai Lee, Frank Raymond. p. cm. — (Mathematical surveys and monographs : v. 166) Includes bibliographical references and index. ISBN 978-0-8218-5231-6 (alk. paper) 1. Fiberings (Mathematics). 2. Complex manifolds. I. Raymond, Frank, 1932– II. Title. QA612.6 .L44 2010 514.2—22 2010022528 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA.
    [Show full text]
  • Topological Spherical Space Forms 3
    TOPOLOGICAL SPHERICAL SPACE FORMS IAN HAMBLETON Abstract. Free actions of finite groups on spheres give rise to topological spherical space forms. The existence and classification problems for space forms have a long history in the geometry and topology of manifolds. In this article, we present a survey of some of the main results and a guide to the literature. 1. Introduction A Clifford-Klein space form is a closed Riemannian manifold with constant sectional curvature. A classical result in differential geometry states that space forms are just the quotients of the standard models Sn, Rn and Hn for spherical, Euclidean, or hyperbolic geometry by discrete groups of isometries acting freely and properly discontinuously (see [21, Chap. 8]). The spherical space forms have now been completely classified, answering the original question of Killing [37] from 1890. Extensive information for both compact and non-compact space forms is contained in Wolf [35]. In 1926, Hopf [36] determined the fundamental groups of three-dimensional spherical space forms, and asked whether every closed, oriented 3-manifold with finite fundamental group admits a Riemannian metric of constant positive curvature. In particular, this topological question is a generalization of the famous Poincar´econjecture to the non- simply connected case. By 2005, both Hopf’s question and the Poincar´econjecture were finally answered by Perelman [8], building on the Ricci flow techniques of Hamilton [54], in the process of veriflying the Thurston Geometrization conjecture (see [17, 25, 29, 30]). The topological spherical space form problem [13] generalizes Hopf’s question to higher dimensions: which finite groups can act freely on spheres via homeomorphisms or diffeo- arXiv:1412.8187v1 [math.GT] 28 Dec 2014 morphisms ? How can one classify free (smooth or topological) finite group actions on spheres ? These questions have led to an enormous amount of research in algebraic and geometric topology over the last hundred years, and substantial progress has been made.
    [Show full text]
  • Qd(P)-FREE RANK TWO FINITE GROUPS ACT FREELY on a HOMOTOPY PRODUCT of TWO SPHERES
    Qd(p)-FREE RANK TWO FINITE GROUPS ACT FREELY ON A HOMOTOPY PRODUCT OF TWO SPHERES MICHAEL A. JACKSON Abstract. A classic result of Swan states that a finite group G acts freely on a finite homotopy sphere if and only if every abelian subgroup of G is cyclic. Following this result, Benson and Carlson conjectured that a finite group G acts freely on a finite complex with the homotopy type of n spheres if the rank of G is less than or equal to n. Recently, Adem and Smith have shown that every rank two finite p-group acts freely on a finite complex with the homotopy type of two spheres. In this paper we will make further progress, showing that rank two groups which are Qd(p)-free act freely on a finite homotopy product of two spheres. 2000 MSC: 57Q91, 55R25, 20C15, 20E15 1. Introduction Recall that the p-rank of a finite group G, rkp(G), is the largest rank of an elementary abelian p-subgroup of G and that the rank of a finite group G, rk(G), is the maximum of rkp(G) taken over all primes p. In addition we can define the homotopy rank of a finite group G, h(G), to be the minimal integer k such that G acts freely on a finite CW-complex Y ' Sn1 ×Sn2 ×· · ·×Snk . Benson and Carlson [7] state a conjecture that for any finite group G, rk(G) = h(G). The case of this conjecture when G is a rank one group is a direct result of Swan’s theorem [28].
    [Show full text]