CONTEMPORARY MATHEMATICS

78

Braids Proceedings of the AMS-IMS-SIAM Joint Summer

Research Conference on Artin's Braid Group held July 13-26. 1986 at the University of California ,

Santa Cruz, California

Joan S. Birman Anatoly Libgober

Editors

Recent Titles in This Series

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Braids

CoNTEMPORARY MATHEMATICS

78

Braids

Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Artin's Braid Group

held July 13-26, 1986 at the University of California, Santa Cruz, California

Joan S. Birman Anatoly Libgober

Editors

American Mathematical Society Providence, Rhode Island

EDITORIAL BOARD

Irwin Kra, managing editor M. Salah Baouendi William H. Jaco Daniel M. Burns David Eisenbud Jonathan Goodman

Gerald J. Janusz Jan Mycielski

The AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on Artin's Braid Group was held at the University of California, Santa Cruz, California on July 13-26, 1986, with support from the National Science Foundation, Grant DMS-8415201.

1980 Mathematics Subject Classification (1985 Revision). Primary 55P, 55S, 57M, 58F, 14B, 46L10, 46L35, 11R29; Secondary 14E20, 14H30, 32B30. 55Q52.

Library of Congress Cataloging-in-Publication Data

AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on Artin's Braid Group (1986: University of California, Santa Cruz)

Braids: proceedings of the AMS-IMS-SIAM joint summer research conference/Joan S. Birman and Anatoly Libgober, editors.

p. cm.-(Contemporary mathematics, ISSN 0271-4132; v. 78) "AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on

Artin's Braid Group ... held at the University of California, Santa Cruz, California on July 13-26, 1986, with support from the National Science Foundation"-T.p. verso.

Includes bibliographical references. ISBN 0-8218-5088-1 (alk. paper) 1. Braid theory-Congresses. I. Birman, Joan S., 1927-11. Libgober, A. (Anatoly),

1949- . III. American Mathematical Society. IV. Institute of Mathematical Statistics. V. Society for Industrial and Applied Mathematics. IV. Title. VII. Series: Contemporary mathematics (American Mathematical Society); v. 78. 5141.224-dc19 88-26283

CIP

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This volume was printed directly from author-prepared copy. The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability. § 10 9 8 7 6 54 3 02 01 00 99 98

V. Jones C. Safont and J. Birman

S. Gitler and A. Libgober

M. Lozano and P. Wong J. Przytycki (V. S. Sunder and A. Ocneanu in background) and J. Kania-Bortoszynska

L. Taylor and F. Cohen J. Franks

B. Wajnryb and B. Moishezon W. Browder

K. Aomoto H. Morton and J. Harer

J. Menasco N. Cozarelli and S. Spengler

J. Harper and D. Sumners B. Jiang

J. Simon and R. Randell

CONTENTS

Organizing Committee

List of Participants

Introduction

A construction of integrable differential system associated with braid groups

K. Aomoto

Mapping class groups of surfaces

XV

XV

xxiii

Joan S. Birman 13

Automorphic sets and braids and singularities E. Brieskorn 45

The operator algebras of the two dimensional Ising model Alan L. Carey and David E. Evans 117

Artin's braid groups, classical homotopy theory, and sundry other curiosities

F. R. Cohen 167

Classification of solvorbifolds in dimension three - I William D. Dunbar 207

Pure braid groups and products of free groups Michael Falk and Richard Randell 217

Polynomial covering maps Vagn Lundsgaard Hansen 229

xi

xii CONTENTS

Arithmetic analogues of braid groups and Galois represcn ta tions

Yasutaka Ihara 245

Application of braids to fixed points of surface maps Boju Jiang 259

Statistical mechanics and the Jones polynomial Louis H. Kauffman 263

Hurwitz action and finite quotients of braid groups Paul Kluitmann 299

Heights of simple loops and pseudo-Anosov homeomorphisms Tsuyoshi Kobayashi 327

Linear representations of braid groups and classical Yang-Baxter equations

Toshitake Kohno 339

A survey of Heeke algebras and the Artin braid groups G. I. Lehrer 365

On divisibility properties of braids associated with algebraic curves

A. Libgober 387

The panorama of polynomials for knots, links and skeins W. B. R. Lickorish 399

The structure of deleted symmetric products R. James Milgram and Peter Loffler 415

Braid group technique in complex geometry, I: Line arrangements in ([.p 2

B. Moishezon and M. Teicher 425

Problems H. R. Morton 557

Polynomials from braids H. R. Morton 575

The Jones polynomial of satellite links about mutants H. R. Morton and P. Traczyk 587

CONTENTS xiii

On the deformation of certain type of algebraic varieties Mutsuo Oka 593

Braids and discriminants Peter Orlik and Louis Solomon 605

tk moves on links Jozef H. Przytycki 615

Mutually braided open books and new invariants of fibered links

Lee Rudolph 657

Generalized braid groups and self-energy Feynman integrals Mario Salvetti , 675

Markov classes in certain finite symplectic representations of braid groups

Bronislaw Wajnryb 687

The braid index of an algebraic link R. F. Williams 697

Markov algebras David N. Yetter 705

ORGANIZING COMMITTEE

J. Birman Ralph Cohen

A. Libgober

J. Franks V. Jones

LIST OF PARTICIPANTS

Roger Alperin Department of Mathematics University of Oklahoma Norman, OK 73069

K. Aomoto Faculty of Science Nagoya University Furo Cho Nagoya 464, Japan

B. Mitchell Baker Department of Mathematics University of Ottawa Ottawa, Ontario, Canada

David W. Barnette Department of Mathematics University of California Davis, CA 95616

William E. Baxter Department of Mathematics University of California Berkeley, CA 94720

XV

Martin Bcndersky Department of Mathematics Rider College Lawrenceville, NJ 08648

Joan S. Birman Department of Mathematics Columbia University New York, NY 10027

R. P. Boyer Department of Mathematics Drexel University Philadelphia, PA 19104

Steven P. Boyer Department of Mathematics University of Toronto Toronto, Ontario MSS 141 Canada

Egbert Brieskorn Mathematics Institute University of Bonn Wegclerstrasse 10 53 Bonn Federal Republic of Germany

xvi LIST OF PARTICIPANTS

William Browder Department of Mathematics Princeton University Washington Road Princeton, NJ 08544

Edgar H. Brown Department of Mathematics Brandeis University Waltham, MA 02254

S. Bullett Department of Mathematics Queen Mary College Mile End Road London E l 4NS, England

Robert Campbell Department of Mathematics University of California Berkeley, CA 94720

Joe Christy Department of Mathematics Northwestern University Evanston, IL 60201

Tim D. Cochran Department of Mathematics University of California Berkeley, CA 94720

Frederick R. Cohen Department of Mathematics University of Kentucky Lexington, K Y 40506

Ralph Cohen Department of Mathematics Stanford University Stanford, CA 94305

Antonio Costa Matern a ticas U ni versidad Compl u tense

de Madrid Ciudad Universitaria Madrid, Spain

Nicholas R. Cozzarelli Department of Molecular

Biology University of California Berkeley, CA 94705

Donco Dimovski Ma tema ticki Insti tu t Prirodno Matematicki

Fakultet 91000 Skopje, Yugoslavia

Claus Ernst Department of Mathematics

and Computer Science Florida State University Tallahassee, FL 32306

D. Evans Mathematics Institute University of Warwick Coventry CU4 7 AL, England

John M. Franks Department of Mathematics Northwestern University Evanston, IL 60201

Richard M. Gillette Department of Mathematical

Sciences Montana State University Bozeman, MT 5971 7

Samuel Gitler Department of Mathematics APDO Postal 14740 Centro de Investigacion, JPN Mexico City, Mexico 14

LIST OF PARTICIPANTS xvii

Fred Goodman Department of Mathematics University of Iowa Iowa City, lA 52242

Nathan Habegger Department of Mathematics U ni versi ty of California

at San Diego La Jolla, CA 92093

Vagn Lundsgaard Hansen Math Insti tu t Danish Tech. University DK 2800 Lyngby, Denmark

J. Harer Department of Mathematics University of Rochester Rochester, NY 14627

John R. Harper Department of Mathematics University of Rochester Rochester, NY 14627

Erika Hironaka Department of Mathematics Brown University P. 0. Box 1917 Providence, RI 02912

Yasutaka Ihara Department of Mathematics University of Tokyo Bunkyo-ku, Tokyo 113, Japan

Norio Iwase Department of Mathematics Kyushu University Fukuoka, Japan 812

Baja Jiang c/o Professor Albrecht Dold Math Institute University of Heidelberg West Germany

Vaughan Jones Department of Mathematics University of California Berkeley, CA 94720

A. Juhasz Department of Pure

Mathematics Weizmann Institute of Science Rehovot 7 6100, Israel

Taizo Kanenobu Department of Mathematics Kyushu University 33 Fukuoka 812, Japan

M. Kania-Bortoszynska Department of Mathematics University of California Berkeley, CA 94720

Mitsutoshi Kato Department of Mathematics Faculty of Science Kyushu University 33 Fukuoka Postal No. 812 Fukuoka, Japan

L. Kauffman Department of Mathematics University of Illinois Chicago, IL 52242

Mark E. Kidwell Department of Mathematics U.S. Naval Academy Annapolis, MD 21402

xviii LIST OF PARTICIPANTS

Robion C. Kirby Department of Mathematics University of California Berkeley, CA 94720

P. Kluitman Ma thema tisches Insti tut Universitat Bonn Beringstr. 4 D-5300 Bonn Federal Republic of Germany

Kazuaku Kobayashi Department of Arts and Sciences Tokyo Women's Christian

University Tokyo 167, Japan

Tsuyoshi Kobayashi Department of Mathematics Osaka University Toyonaka, Osaka 560, Japan

Toshitake Kohno Department of Mathematics Nagoya University Nagoya 464, Japan

Hideki Kosaki Department of Mathematics College of Genera I Education Kyushu University Fukuoka, 810, Japan

Nicholas Kuhn Department of Mathematics Princeton University Princeton, NJ 08544

Le Dung Trang Centre de Mathematiques Ecole Polytechnique 91128 Palaiseau Cedex, France

Gustav I. Lehrer Department of Mathematics University of Sydney Sydney, NSW 2006, Australia

Anatoly S. Libgobcr Department of Mathcma tics U ni versi ty of Illinois Chicago, IL 60680

W. B. Raymond Lickerish Department of Pure

Mathematics Cambridge University 16 Mill Lane Cambridge, CB2 15B, England

David D. Long Department of Mathematics University of California Santa Barbara, CA 93106

Roberto Longo Department of Mathematics University of Roma La Sapienze Piazzale A. Moro 2 00185 Roma, Italy

M. Lozano Department of Geometry and

Topology University of Zaragoza Zaragoza, 50009, Spain

Yoshihiko Marumoto Department of Mathematics Faculty of Education Saga University Saga 840, Japan

J. Peter May Department of Mathematics University of Chicago 5734 S. University Avenue Chicago, IL 6063 7

LIST OF PARTICIPANTS xix

Curtis McMullen Institute for Advanced Study Princeton, NJ 08540

William Wyatt Menasco Department of Mathematics SUNY at Buffalo Buffalo, NY 14222

R. J. Milgram Department of Mathematics Stanford University Stanford, CA 94305

Kenneth C. Millett Department of Mathematics U ni versi ty of California Santa Barbara, CA 93106

Mamoru Mimura Department of Mathematics Okayama University Okayama 700, Japan

Tadayoshi Mizutani Department of Mathematics University of California Berkeley, CA 94720

Boris Moishezon Department of Mathematics Columbia University New York, NY 10027

Hugh R. Morton Department of Mathematics University of Liverpool Liverpool L69 3BH, England

Hitoshi Murakami Department of Mathematics Osaka City University Sumiyoshi-hu Osaka 558, Japan

Kunio Murasugi Department of Mathematics University of Toronto Toronto, Ontario M5S !AI Canada

Adrian Ocneanu Mathematical Sciences

Research Institute 100 Centennial Dr. Berkeley, CA 94720

Ronald S. Ojakian Department of Biophysics University of California Berkeley, CA 94720

Mutsuo Oka Tokyo Institute of Technology Department of Mathematics Faculty of Science OH-Okayama, Meguro-Ku Tokyo, Japan

Peter P. Orlik Department of Mathematics University of Wisconsin Madison, WI 53706

David John Pengelley Department of Mathematics New Mexico State University Las Cruces, NM 88003

J ozcf H. Przytycki Department of Mathematics Universytet Warszawski Palac Kultury i Nauki, P. IH 00900 Warsza wa, Poland

Richard C. Randell Department of Mathematics University of Iowa Iowa City, IA 52242

XX LIST OF PARTICIPANTS

Frank S. Rimlinger Department of Mathematics Columbia University New York, NY 10027

Mark W. Rinker Department of Mathematics U ni versi ty of California Berkeley, CA 94720

Lee N. Rudolph P.O. Box 251 Adamsville, RI 0280 I

Carmen Safont Department of Mathematics Universidad de Zaragoza 50009 Zaragoza, Spain

Kyoji Saito Department of Mathematics,

RIMS Kyoto University Kyoto 606, Japan

Mario Salvetti Departimento di Matematica Via Buonarroti, 2 56100 Pisa, Italy

Noriko M. Sasano Department of Mathematics Tsuda College Kodaira, Tokyo, 187, Japan

Kazuhiro Sasano Department of Mathematics Toyama Medical and Pharma-

ceutical University 2630 Sugitani Toyama, Toyama 930-01 Japan

Martin Scharlemann Department of Mathematics University of California Santa Barbara, CA 93106

Jonathan K. Simon Department of Mathematics University of Iowa Iowa City, lA 52242

Richard Skora Department of Mathematics Indiana U ni versi ty Bloomington, IN 47405

John C. Sligar Department of Mathematics University of Georgia Athens, GA 30604

Edwin Spanier Department of Mathematics University of California Berkeley, CA 94720

Sylvia J. Spengler Biomed Division Lawrence Berkeley Lab. Berkeley, CA 94720

Neal Stolfzfus Department of Mathematics Louisiana State University Baton Rouge, LA 70803

DeWitt Sumners Department of Mathematics Florida State University Tallahassee, FL 32306

V. S. Sunder Department of Mathematics Indian Statistical Institute New Delhi, 110016, India

Laurence R. Taylor Department of Mathematics University of Notre Dame Notre Dame, IN 46556

LIST OF PARTICIPANTS xxi

Michishige Tezuka Department of Mathematics Institute of Tokyo Technology Tokyo, Japan

Pa wel Traczyk Department of Mathematics University of Liverpool Liverpool L69 3BH, England

Jim Van Buskirk Department of Mathematics University of Oregon Eugene, OR 97403

Bronislaw Wajnryb Technion Israel Institute of Technology 32000 Haifa, Israel

Hans Wenzl Department of Mathematics University of California Berkeley, CA 94720

Wilbur Whitten School of Mathematics Institute for Advanced Study Princeton, NJ 08540

Robert F. Williams Department of Mathematics North western U ni versi ty Evanston, IL 60201

Peter N. Wong Department of Mathematics U ni versi ty of Wisconsin Madison, WI 53706

Nobuaki Yagita Department of Mathematics Musashi Institute of Technology Setagoya, Tokyo 158, Japan

Koichi Yano Department of Mathematics Kyushu University Fukuoka 812, Japan

David N. Yetter Department of Mathematics Clark University Worcester, MA 01610

INTRODUCTION

Braid groups were introduced into the mathematical

literature in 1925 in a seminar paper by E. Artinl), although the

idea was implicit in Hurwitz's 1891 manuscript2 ). In the years

since, and particularly in the last 5-10 years, they have played

a role in diverse and unexpected ways in widely different areas

of mathematics, including knot theory, homotopy theory, singu-

larity theory, dynamical systems, and most recently operator

algebras, where exciting new discoveries are closing the gap by

having striking applications to knots and links. This volume

contains the Proceedings of a conference on BRAIDS which was

held in Santa Cruz, california during July, 1986, Its purpose

was to bring together specialists from these different areas of

mathematics, so that they could discuss their discoveries and

exchange ideas and open problems concerning this important and

fundamental group. The conference was truly interdisciplinary.

Intuitively, a braid.is the following object: take two bars

(a top one and a bottom one), each with n hooks attached,

equally spaced along the bars. Join the top bar to the bottom

bar by n strings, inducing a permutation of the hooks and a

weaving pattern in the strings. A typical braid might look like

those pictured in Figure l. Braids are composed by placing one

under the other and deleting the middle bar. It's not hard to

see that inverses exist, and that one has a group; it is the

l) Artin, E., Theorie der Zopfe,Hamb Abh. 4(1925), p.47-72. 2) Hurwitz, A., "Uber Riemannsch. Flachen mit gegebenan

Verzweigungspunhten", Math. Ann. 39, p. l-61.

xxiii

xxiv INTRODUCTION

non-trivial 4-braid identity 4-braid

Figure l

classical braid group Bn. More precisely, let Xn be the quotient space of En-diagonal, n = 1,2, ... , under the natural action of the symmetric group (permuting coordinates). The braid group

Bn is ~ 1 xn. It maps homomorphically onto the symmetric group Sn in an obvious way.

Here are some brief descriptions of the ways in which braids enter into different areas of mathematics.

A. Knot Theory. Artin introduced his group with the idea that braids might be useful in the study of knots and links. If one identifies the top and bottom of each braid string one ob-tains a closed one-manifold which inherits (from the way that the braid is embedded in a3 ) a natural embedding in a3 It was proved by Alexander1 ) that every knot or link may be so-repre-sented, in many ways. The equivalence relation in the various braids which define a given knot or link type was discovered by Markov in 19352 ): it is a union of conjugacy classes in these-quence of braid groups B , B 1 ,B 2 , ... , where n0 is the no no+ no+ braid index of the link in question. A representation

-1 (n-1) x (n-1) matrices over Z[t,t ) of B by n

was discovered by Burau3 >, and using it one may compute the Alexander invariants of knots

1) Alexander, J.W. "A Lemma on systems of knotted curves", Proc, Nat. Acad. Sci. USA 9 (1923), 93-95.

2) Markov, A.A., "Ub~r die freie Aquivalenz geschlossener Zopfe", Recueil Mat Mosco ~(1935, 73-78.

3) Burau, "Uber zopfgruppen und gleichsinnig verdrillte Verke Abh Math Sem Hanischen Univ 11 (1936), 171-178.

INTRODUCTION XXV

outstanding open problem.)

Markov's equivalence relation seemed pretty intractible

until 1969, when Garside succeeded in solving the conjugacy pro-

blem in the braid group1 >. (His solution was soon generalized by Breiskorn and Saito2 ), who discovered important connections

between reflection groups and braids. See also Deligne's work3 ).

The Birman monograph4 ) appeared in 1974, and it contained a

problem list related to the possibility of studying knots and

links via braids. During the years 1974-1983 there was some

progress5 ), but it could not be said that braids were an essen-

tial tool for the study of knots and links. Then, in June 1984,

everything changed with Jones' discovery of a remarkable new

polynomial invariant of knots and links6 ). Jones' invariant is

a trace function on certain C*-algebras. His algebras have a

matrix representation, which includes in its units a represen-

tation of B . n The trace is invariant on Markov's equivalence

classes. The implications of Jones' discoveries are bound to

have a fundamental impact on knots and links and 3-manifolds, as

they become better understood.

Braid groups also enter into the theory of surface mappings.

One way in which this occurs is that Dehn twists about two loops

which intersect once play the role of elementary braids which

have a common string.

B. Singularity theory and reflection groups. Braid groups

were recognized as fundamental groups of the spaces of complex

polynomials of fixed degree without multiple roots very

l) Garside,F.A., "The braid group a·nd other groups", Quart. J. Math. Oxford 20, No. 78(1969), 235-254.

2) Brieskorn,E.,and Saito, K., Artin Gruppen and coxeter gruppen. Inv. Math. 17 (1972), 245-271.

3) Deligne, P., "Les immeubles des groupes de tresses gen-eralises", Inv. Math .l.?. (1972), 273-302

4) J. Birman, Braids, links and mapping class groups, Annals of Math. Studies No. 82, Princeton Univ. Press, 1974.

5) Bennequin, D., "Entrelacements et equations de Pfaff", These de Doctorat d'Etat, Unlversite de Paris VII, Nov. 1982.

6) Jones, Vaughn F.R., "A new polynomial invariant for knots and links", preprint.

xxvi INTRODUCTION

early. (See ref. 1) on p. 1.) These spaces are the complements to the discriminants in the bases of semi-universal deformations

n of the singularities y = x . It was realized by V. Arnold and E. Brieskorn that the fundamental groups of the complement of discriminats of semi-universal deformations of other singular-

ities are similar to the Artin braid groups and encode important

information on these singularities. Brieskorn considered so

called simple singularities corresponding to the simple root

system An' Dn' E6 , E7 , E8 and found presentations of the corres-ponding fundamental groups. 1 ) Jointly with K. Saito he solved the word and conjugacy problems in these "Brieskorn braid groups" 2 ) (corresponding to other simple Lie groups as well). This was also done by P. Deligne3 ). The cohomology of the braid

groups was crucial in Arnold's work on the thirteenth Hilbert

problem4 ). On the other hand the cohomology of pure (colored)

Artin and Brieskorn braid groups have beautiful relationships with the Weyl groups associated with the corresponding root sys-tems. The work prior to 1970, was surveyed by Brieskorn in his report in Seminaire Bourbaki5 ). Since then this line of research has developed very rapidly. Brieskorn's results on deformation

of simple singularities were extended by Looijenga and others to 1) Brieskorn, E., Singular elements of Semisimple algebraic

groups. Actes Cong. Int. Math. Nice 1970. Gauthier-Villars. Paris 1971.

Brieskorn, E., Die Fundamentalgruppe des Raumes der Regularen Obrits einer endlichten komplexen Spiegelungs gruppe. Inv. Math. 12 (1971), 57-61.

2) See 2) on page 3.

3) See 3) on page 3,

4) Arnold, v., Topological invariants of algebraic func-tions II. Funct. Anal. Appl. 4 (1970), p. 91-98.

Arnold, V., Cohomology Classes of Algebraic functions invariant under Tschirnhausen Transformations. Funct. Anal. Appl. 4 (1970), p. 74-75.

5) Brieskorn, E., Surles groupes des tresses [d'Aprer Arnold]. Sem. Bourbaki 1971/1972 No.4 of Lectures Notes, vol. 317.

INTRODUCTION xxvii

simply elliptic and cusp singularities 1 ). Presentations for the

corresponding fundamental groups were found by Van der Lek as

extended Artin groups 2 ). Results on the relationship between

the cohomology of pure braid groups and Weyl groups was ex-

tended by Orlik, solomon and Terao to other Coxeter groups and 3) even arrangements of hyperplanes . Nevertheless many open

problems remain (e.g. , 4 ) Problem 17, and S) Question 8).

Braid groups appeared quite independently in another school

of thought as part of the global study of singularities. In the

1930's 0. Zariski6 ) constructed hypersurfaces in ~pn for which the fundamental group of the complements are groups closely

related to the braid groups of oriented surfaces of arbitrary

genus. He found presentations of these groups and braid groups

of Riemann surfaces (this work was completed by Kaneko7 >). Braid groups played an important role in virtually forgotten

works of the Italian school. (See b) and the extensive biblio-

graphy there). B. Moishezon9 ) approached the problem of

1) Looijenga, E., Homogeneous spaces associated to certain semiuniversal deformations. Proc. Int. cong. Math. Helsinki, 1980, vol 2, p. 529-536.

Looijenga, E., Rational surfaces with an anticanonical cycle. Ann. of Math., 1981, vol. 114, p. 267-322.

2) H. van der Lek, Extended Artin groups. Proc. Symp. in Pure Math. vol. 40, part 2, p. 117-122. AMS 1983.

3) Orlik, P., and Solomon, L., coxeter-Arrangements. Proc. Symp. in Pure Math., vol. 40, part 2, p. 269-291. 1983.

4} Arnold, V., Some open problems in the theory of singu-larities, Proc. Aymp. in Pure Math. vol. 40, Part 1, (1983), p. 57-70.

5) Le-Tessier, Report on the problem session. Proc. Symp. in Pure Math. vol. 40, part 2, p. 105-116. AMS 1983.

6) Zariski, 0., "On the Poincare group of rational plane curve", Amer. J. Math. 58 (1936), 607-619.

Zariski, 0., "The topological discriminant group of a Riemann surface of genus p. Amer. J. Math. (1937), 335-358.

Dolgachev-Libgober, On the fundamental group of the com-plement to a discriminant variety. Lecture Notes in Math., vol. 862, pp. 1-25, Springer-Verlag, 1981.

7) Kaneko. Preprint, Kyushi University. 8) Chisini, 0. courbes de diramation des planes multiple

et tresses algebriques. Deuxieme Colloque de Geometrie Alge-brique tenu a Liege les 9,10,11 ef 12 June 1952, CBRM, 11-27.

9) Moishezon, B., Stable branch curves and braid monodromies, Lecture Notes in Math., vol.862, 107-192. Springer-Verlag, 1981.

xxviii INTRODUCTION

classification of algebraic surfaces of general type by repre-

senting them as branched covers of ~E 2 and describing surfaces using branching locus. These loci were studied by Moishezon in

terms of braid monodromy which he views as a factorization of

the generator of the center of the braid group. Many questions

in algebraic geometry such as structure of various fundamental

groups, degenerations, moduli space, homotopy type can be trans-

lated into combinatorial quest1ons about braid groups 5). The

topology of algebraic curves was also studied using braids by

L. Rudolph2 ).

Automorphism groups of braid groups are important for the

study of families of plane curves with singularities3 ).

Brieskorn braid groups appear in these global problems as well4 ),

but here only the first steps have been taken.

In the study of differential equations with regular sin-

gularities (notably in the study of hypergeometric equations)

the role of the braid group as the fundamental group of the

complement to a discriminant was apparent for some time,

(see Aomoto's work in this volume and references there). Re-

cently the seminal work of H. A. Schwartz on the monodromy

group of hypergeometric equations was reconsidered and gen-

eralized to higher dimensions (first work on these general-

izations can be traced back to E. Picard) with Braid

groups playing a fundamental role. (G. Mostow, Bull. AMS

l) Moishezon, B., Algebraic surfaces and arithmetics of braids. Progress in Math., vol. 36, Burkhauser 1983.

Libgober, A., On the homotopy type of the complement to Plane algebraic curves. Journ. fur die reine und ang. Math. Band 367 p. 103-114, 1986.

2) Rudolph, L., "Some knot theory of complex plane curves" L'Ens. Math. t. 29 (1983), p. 185-208.

3) Artin, M., Masur, B., Introduction p. 8-9 in Collected papers by 0. Zariski. MIT Press, 1978.

4) Libgober, A., On the fundamental group of the space of cubic surfaces, Math. Zeit. 162 (1978), p. 63-67.

5) Fundamental groups of the complements to plane singular curves. Proc. of Symp. in Pure Math. 46, Summer Institute on Algebraic Geometry. Bowdin, College, Maine, pp. 29-45 (1988).

INTRODUCTION xxix

vol.l6 No. 2 and references there.) C. Homotopy Theory. The relevance of Artin's braid group

to homotopy theory first became apparent in the early 1970's in studies made by J.P. May and F.R. Cohen1) of the combinatorial

and algebraic structure of iterated loop spaces. Roughly speak-

ing, they found that the deep relation between the homology of

the symmetric group and the structure of the stable homotopy

groups of spheres as studied by Dyer and Lashoff, Quillen and others has an unstable analogue, namely there is a direct and

deep relationship between the homology of B and the homotopy n type of the 2-dimensional sphere s2 . Furthermore, the stabili-zation process is seen on the group-theoretic level via the homomorphism from B to S given by sending a braid to the per-n n mutation of the end points of the strings. These results were

exploited by F. Cohen and L. Taylor to give the first calcula-tion of the homology of the pure braid group2 ). They were also used by M. Mahowald3 ) and R. cohen4 ) in the construction of infinite families in the homotopy groups of spheres. Also, they were an essential ingredient in the work of Brown and Peterson5) and of R. cohen6 ) which resulted in a proof of the conjecture that every compact n-manifold immerses in R2n-a(n), where a(n)

is the number of l's in the dyadic expansion of n.

1) May, J.P., The geometry of iterated loop spaces, Springer Lecture Notes No. 271, 1972.

cohen, F.R., Lada, T.J., and May, J.P., The homology of iterated loop spaces, Springer Lectures Notes No. 533, 1976.

2) cohen, F.R., and Taylor, L.R., computations of Gelfand-Fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, Springer Lecture Notes No. 657, pp. 106-143, 1978.

3) Mahowald, M., A new infinite family in 2n:, Topology 16 (1977), 249-256.

4) Cohen, R.L., Odd Primary infinite families in stable homotopy theory, Memoirs of A.M.S. 242(1981).

5) Brown, E.H. and Peterson, F.P., A universal space for normal bundles of n-manifolds, comment. Math. Helv. 54 (1979) 405-430.

6) cohen, R.L., The immersion conjecture for differentiable manifolds, to appear.

XXX INTRODUCTION

Ongoing research in homotopy theory that is making use of

the braid groups and their relation to the 2-sphere include the

work of P. Goerss, J. Jones and M. Mahowald, in which they seek

to apply braid group theory to both algebraic and geometric

K-theory.

D. C* Algebras. Here braids are very new, and also very

exciting. The first explicit reference to the braid groups in . 1 b . l) h b relat~on to operator a ge ra appears ~n w ere Jones o -

tained an interesting one-parameter family of representations

of Bn while solving a problem on type II 1 factors. If N s M are

rings with the same identity, one may define a notion of index

[M:N]. Jones showed in1 ) that for II 1 factors [N:M] is any real

number 2 4 or one of the number 4 cos2 n/n, n = 3,4,5 .... His proof of this result introduces a tower which can be thought of

as being tied up by the braid groups. The tower is defined

inductively by M0 = N, M1 = M, and Mi =End (M.), M. being M. 1 ~ ~ ~-

a right M. 1-module. ~- Note that Mi-ls Mi. Inside Mi is the If one makes orthogonal projection onto Mi-l which we call e ..

-1 ~ the change of variables [M:N] = 2 + t + t , g. = (t+l)e. - 1,

~ ~

one sees that the g. satisfy the braid group relations, and thus ~

one obtains representations (not necessarily unitary) of the

groups Bn. A further analysis of the algebras generated by the g.'s

~

reveals that braid groups were at least implicit in some pre-

vious related works, notably Temperley and Lieb's analysis of

the Potts model in statistical mechanics2 ), onsager's solution

of the Ising model, Powers' construction of type III factors

(where the braid group can be used to prove factoriality3 ),

Chutz's algebras 0 (which form a universal object for all the n

1) Jones, V.F.R., "Braid groups, Heeke algebras and type II, factors", to appear in Proceeding Japan-US converence 1983.

2) Temperley and Lieb, Proc. Royal Soc. London (1971), 251-280.

3) Powers, R.T., "Representations of uniformly hyperfinite algebras and their associated von Neumann algebras", Ann. Math. 86 (1967), 38-171.

INTRODUCTION xxxi

algebra going on) and more recently a paper by Pimsner and popa 1 . h . d t' l) re at~ng entropy to t e ~n ex ques ~on .

Perhaps the most striking fact is a recent observation of

Jones that the IIl factor trace that is present in all these

algebras allows one to define a new polynomial invariant for

knots and links. This invariant seems quite powerful and has

already settled some problems in knot theory. The use of braids

in C*-algebras and statistical mechanics can be expected to

increase dramatically as our understanding of the relationship

between the above works deepens. In particular it seems likely

that one will be able to say new things about the q-state Potts

model by analyzing the representations described above.

E. Dynamical systems. closed orbits in a flow on R3 or s 3 or other 3-manifolds have knot and link types, and since the flow

has a natural period near the orbit, the orbit can often be view-

ed as a braid about some axis (e.g. another orbit). The first

observation that the knots which arise in a flow ought to form

a class of related knots appear to have been in2 ). In 3 ) J.

Franks showed that there can be a close relationship between the

symbolic dynamics of non-singular Smale flows on s 3 and the Alexander invariants of the link of closed orbits. The class of

links determined by the closed orbits in Lorenz's equations were

studied by Birman and Williams4 ): they turn out to be closed

braids, in fact a class of braids which yields a new and inter-

esting class of knots. See also4 ). The connection between

braids and period doubling in certain suspension flows is the

l) pimser M., and papaS., "Entropy and index for subfac-tors", Preprint INCREST, Bucharest, Romania (1983).

2) Morgan, John, "Non-singular Marse-Smale flows on 3-di-· ensional manifolds", Topology 18 (1978), 41-53.

3) Franks, John, "Knots,1inks and symbolic dynamics", Annals of Math. 113 (1981), 529-552.

4) Birman,---::T. and williams R., "Knotted periodic orbits in dynamical systems I: Lorenz's equations", Topology 22 (No. l) 47-82, 1983.

5) Williams, R.F., "Lorenz knots are prime", preprint.

xxxii INTRODUCTION

d '11' l) subject of new work by Holmes an W1 1ams . Braiding also

plays a role in Handell's ongoing research on surface mappings,

and in the new results of Boyland2 ). This is a rapidly develop-

ing area of research.

F. Fixed point theory. Recent work of Jiang3 ), and of

Fadell and Husseini4 ) use braids to exhibit obstructions to

deforming a map on a manifold M to one with a minimum number of

fixed points.

G. Number theory. New and unexpected use of braids was

initiated by Y. Ihara. He introduced a profinite analog of

the braid groups considering automorphisms of free pro-t-groups.

The Galois group of the algebraic closure of rationals has

natural homomorphisms into these braid groups. Interplay of

these different groups lead him to connections with Jacobi sums,

Vandiver conjecture etc. See his introduction to his work in

this volume.

H. complexity. A few months after the BRAIDS conference

we were interested to learn of new applications of braid theory

to complexity theory. Let Pd be the space of n-tuples of com-

plex numbers, regarded as the coefficient space of all monic d d' polynomials q(z) of degree d. Let E be a copy of p , regarded

now as the root space.

the discriminant in pd,

and ~ 1 (Pd-~) the braid yield a natural map f:

local sections for f.

d E , and ~ If ~ is the diagonal in d then n 1 (E -6) is the pure braid group

group. The elementary synmetric functions

Ed ~ Pd. Root finding algorithms yield

Smale has defined and studied the "topological complexity"

of an arbitrary algorithm to compute the roots of q(z) to

1) Holmes P., and Williams, R.F. "Knotted periodic orbits in the suspension of Smale's horseshow: Torus knots and bifurca-tion sequences", preprint.

2) Boyland, P., "Braid types and a topological method of proving positive entropy", preprint, 1984.

3) Jiang, B., "Fixed points and braids", preprint. 4) Fadell, E. and Husseini, S., "The Nielsen number on

surfaces," contemporary Math. 21 (1983), 59-98.

INTRODUCTION xxxiii

within 1) £ The cohomology of the braid group plays an import-ant role in his work. His interesting theorem is that for all sufficiently small e, the topological complexity is branched

2/3 below by (log2d) . Very recently v. Vasiljev announced an improvement of this bound.

All of these themes are not explored in equal detail in

this volume, which is a mix of expositing articles and new research. we did not attempt to give a coherent presentation

(although several of our contributors do give scholarly, coher-ent reviews of individual areas). Our true hope is that the conference and this volume will stimulate thought and lead to

new mathematics. In closing, we take this opportunity to thank everyone who

contributed to the success of the conference. We thank the

National Science Foundation for financial support. we thank the American Mathematical Society for the administrative help which it provided in all phases of the conference organization. The campus at the University of California Santa cruz was an exceptionally beautiful and tranquil location, with unfailingly cooperative weather.

This introduction was based in large part upon material prepared in 1984 for the proposal to hold the conference. That proposal was written jointly by the five members of the Organ-

izing committee. We thank Vaughan Jones, Ralph Cohen and John

Franks for that, and for all of their help in the planning and running of the conference. Special thanks go to R. cohen for help in preparations of this volume. Finally, we thank the participants. Their enthusiastic participation in the inter-disciplinary spirit of the conference, and their willingness to explain their own work so that non-experts could understand it, made the BRAIDS Conference a memorable occasjon where barriers

between specialists in diverse fields were broken down.

Joan Birman Anatoly Libgober

1) Smale, S., On the topology of algorithms, Journal of Complexity Theory, 3, 1987.

ISBN 0-8218-5088-1

9 780821 850886