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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 http://dx.doi.org/10.1090/conm/078

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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 Gerald J. Janusz David Eisenbud Jan Mycielski Jonathan Goodman 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

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionOams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

Copyright @1988 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. 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 XV

List of Participants XV

Introduction xxiii

A construction of integrable differential system associated with braid groups K. Aomoto

Mapping class groups of surfaces 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 ([.p2 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 J. Franks Ralph Cohen V. Jones A. Libgober

LIST OF PARTICIPANTS

Roger Alperin Martin Bcndersky Department of Mathematics Department of Mathematics University of Oklahoma Rider College Norman, OK 73069 Lawrenceville, NJ 08648

K. Aomoto Joan S. Birman Faculty of Science Department of Mathematics Nagoya University Columbia University Furo Cho New York, NY 10027 Nagoya 464, Japan R. P. Boyer B. Mitchell Baker Department of Mathematics Department of Mathematics Drexel University University of Ottawa Philadelphia, PA 19104 Ottawa, Ontario, Canada Steven P. Boyer David W. Barnette Department of Mathematics Department of Mathematics University of Toronto University of California Toronto, Ontario MSS 141 Davis, CA 95616 Canada

William E. Baxter Egbert Brieskorn Department of Mathematics Mathematics Institute University of California University of Bonn Berkeley, CA 94720 Wegclerstrasse 10 53 Bonn Federal Republic of Germany

XV xvi LIST OF PARTICIPANTS

William Browder Antonio Costa Department of Mathematics Matern a ticas Princeton University U ni versidad Compl u tense Washington Road de Madrid Princeton, NJ 08544 Ciudad Universitaria Madrid, Spain Edgar H. Brown Department of Mathematics Nicholas R. Cozzarelli Brandeis University Department of Molecular Waltham, MA 02254 Biology University of California S. Bullett Berkeley, CA 94705 Department of Mathematics Queen Mary College Donco Dimovski Mile End Road Ma tema ticki Insti tu t London E l 4NS, England Prirodno Matematicki Fakultet Robert Campbell 91000 Skopje, Yugoslavia Department of Mathematics University of California Claus Ernst Berkeley, CA 94720 Department of Mathematics and Computer Science Joe Christy Florida State University Department of Mathematics Tallahassee, FL 32306 Northwestern University Evanston, IL 60201 D. Evans Mathematics Institute Tim D. Cochran University of Warwick Department of Mathematics Coventry CU4 7 AL, England University of California Berkeley, CA 94720 John M. Franks Department of Mathematics Frederick R. Cohen Northwestern University Department of Mathematics Evanston, IL 60201 University of Kentucky Lexington, K Y 40506 Richard M. Gillette Department of Mathematical Ralph Cohen Sciences Department of Mathematics Montana State University Stanford University Bozeman, MT 5971 7 Stanford, CA 94305 Samuel Gitler Department of Mathematics APDO Postal 14740 Centro de Investigacion, JPN Mexico City, Mexico 14 LIST OF PARTICIPANTS xvii

Fred Goodman Baja Jiang Department of Mathematics c/o Professor Albrecht Dold University of Iowa Math Institute Iowa City, lA 52242 University of Heidelberg West Germany Nathan Habegger Department of Mathematics Vaughan Jones U ni versi ty of California Department of Mathematics at San Diego University of California La Jolla, CA 92093 Berkeley, CA 94720

Vagn Lundsgaard Hansen A. Juhasz Math Insti tu t Department of Pure Danish Tech. University Mathematics DK 2800 Lyngby, Denmark Weizmann Institute of Science Rehovot 7 6100, Israel J. Harer Department of Mathematics Taizo Kanenobu University of Rochester Department of Mathematics Rochester, NY 14627 Kyushu University 33 Fukuoka 812, Japan John R. Harper Department of Mathematics M. Kania-Bortoszynska University of Rochester Department of Mathematics Rochester, NY 14627 University of California Berkeley, CA 94720 Erika Hironaka Department of Mathematics Mitsutoshi Kato Brown University Department of Mathematics P. 0. Box 1917 Faculty of Science Providence, RI 02912 Kyushu University 33 Fukuoka Postal No. 812 Yasutaka Ihara Fukuoka, Japan Department of Mathematics University of Tokyo L. Kauffman Bunkyo-ku, Tokyo 113, Japan Department of Mathematics University of Illinois Norio Iwase Chicago, IL 52242 Department of Mathematics Kyushu University Mark E. Kidwell Fukuoka, Japan 812 Department of Mathematics U.S. Naval Academy Annapolis, MD 21402 xviii LIST OF PARTICIPANTS

Robion C. Kirby Gustav I. Lehrer Department of Mathematics Department of Mathematics University of California University of Sydney Berkeley, CA 94720 Sydney, NSW 2006, Australia

P. Kluitman Anatoly S. Libgobcr Ma thema tisches Insti tut Department of Mathcma tics Universitat Bonn U ni versi ty of Illinois Beringstr. 4 Chicago, IL 60680 D-5300 Bonn Federal Republic of Germany W. B. Raymond Lickerish Department of Pure Kazuaku Kobayashi Mathematics Department of Arts and Sciences Cambridge University Tokyo Women's Christian 16 Mill Lane University Cambridge, CB2 15B, England Tokyo 167, Japan David D. Long Tsuyoshi Kobayashi Department of Mathematics Department of Mathematics University of California Osaka University Santa Barbara, CA 93106 Toyonaka, Osaka 560, Japan Roberto Longo Toshitake Kohno Department of Mathematics Department of Mathematics University of Roma Nagoya University La Sapienze Piazzale A. Moro 2 Nagoya 464, Japan 00185 Roma, Italy

Hideki Kosaki M. Lozano Department of Mathematics Department of Geometry and College of Genera I Education Topology Kyushu University University of Zaragoza Fukuoka, 810, Japan Zaragoza, 50009, Spain

Nicholas Kuhn Yoshihiko Marumoto Department of Mathematics Department of Mathematics Princeton University Faculty of Education Princeton, NJ 08544 Saga University Saga 840, Japan Le Dung Trang Centre de Mathematiques J. Peter May Ecole Polytechnique Department of Mathematics 91128 Palaiseau Cedex, France University of Chicago 5734 S. University Avenue Chicago, IL 6063 7 LIST OF PARTICIPANTS xix

Curtis McMullen Kunio Murasugi Institute for Advanced Study Department of Mathematics Princeton, NJ 08540 University of Toronto Toronto, Ontario M5S !AI William Wyatt Menasco Canada Department of Mathematics SUNY at Buffalo Adrian Ocneanu Buffalo, NY 14222 Mathematical Sciences Research Institute R. J. Milgram 100 Centennial Dr. Department of Mathematics Berkeley, CA 94720 Stanford University Stanford, CA 94305 Ronald S. Ojakian Department of Biophysics Kenneth C. Millett University of California Department of Mathematics Berkeley, CA 94720 U ni versi ty of California Santa Barbara, CA 93106 Mutsuo Oka Tokyo Institute of Technology Mamoru Mimura Department of Mathematics Department of Mathematics Faculty of Science Okayama University OH-Okayama, Meguro-Ku Okayama 700, Japan Tokyo, Japan

Tadayoshi Mizutani Peter P. Orlik Department of Mathematics Department of Mathematics University of California University of Wisconsin Berkeley, CA 94720 Madison, WI 53706

Boris Moishezon David John Pengelley Department of Mathematics Department of Mathematics Columbia University New Mexico State University New York, NY 10027 Las Cruces, NM 88003

Hugh R. Morton J ozcf H. Przytycki Department of Mathematics Department of Mathematics University of Liverpool Universytet Warszawski Liverpool L69 3BH, England Palac Kultury i Nauki, P. IH 00900 Warsza wa, Poland Hitoshi Murakami Department of Mathematics Richard C. Randell Osaka City University Department of Mathematics Sumiyoshi-hu University of Iowa Osaka 558, Japan Iowa City, IA 52242 XX LIST OF PARTICIPANTS

Frank S. Rimlinger Jonathan K. Simon Department of Mathematics Department of Mathematics Columbia University University of Iowa New York, NY 10027 Iowa City, lA 52242

Mark W. Rinker Richard Skora Department of Mathematics Department of Mathematics U ni versi ty of California Indiana U ni versi ty Berkeley, CA 94720 Bloomington, IN 47405

Lee N. Rudolph John C. Sligar P.O. Box 251 Department of Mathematics Adamsville, RI 0280 I University of Georgia Athens, GA 30604 Carmen Safont Department of Mathematics Edwin Spanier Universidad de Zaragoza Department of Mathematics 50009 Zaragoza, Spain University of California Berkeley, CA 94720 Kyoji Saito Department of Mathematics, Sylvia J. Spengler RIMS Biomed Division Kyoto University Lawrence Berkeley Lab. Kyoto 606, Japan Berkeley, CA 94720

Mario Salvetti Neal Stolfzfus Departimento di Matematica Department of Mathematics Via Buonarroti, 2 Louisiana State University 56100 Pisa, Italy Baton Rouge, LA 70803

Noriko M. Sasano DeWitt Sumners Department of Mathematics Department of Mathematics Tsuda College Florida State University Kodaira, Tokyo, 187, Japan Tallahassee, FL 32306

Kazuhiro Sasano V. S. Sunder Department of Mathematics Department of Mathematics Toyama Medical and Pharma- Indian Statistical Institute ceutical University New Delhi, 110016, India 2630 Sugitani Toyama, Toyama 930-01 Japan Laurence R. Taylor Department of Mathematics Martin Scharlemann University of Notre Dame Department of Mathematics Notre Dame, IN 46556 University of California Santa Barbara, CA 93106 LIST OF PARTICIPANTS xxi

Michishige Tezuka Robert F. Williams Department of Mathematics Department of Mathematics Institute of Tokyo Technology North western U ni versi ty Tokyo, Japan Evanston, IL 60201

Pa wel Traczyk Peter N. Wong Department of Mathematics Department of Mathematics University of Liverpool U ni versi ty of Wisconsin Liverpool L69 3BH, England Madison, WI 53706

Jim Van Buskirk Nobuaki Yagita Department of Mathematics Department of Mathematics University of Oregon Musashi Institute of Technology Eugene, OR 97403 Setagoya, Tokyo 158, Japan

Bronislaw Wajnryb Koichi Yano Technion Department of Mathematics Israel Institute of Technology Kyushu University 32000 Haifa, Israel Fukuoka 812, Japan

Hans Wenzl David N. Yetter Department of Mathematics Department of Mathematics University of California Clark University Berkeley, CA 94720 Worcester, MA 01610

Wilbur Whitten School of Mathematics Institute for Advanced Study Princeton, NJ 08540 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, singularity 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 obtains 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 ,B , ... , where n is the no no+ 1 no+ 2 0 braid index of the link in question. A representation of B by n -1 (n-1) x (n-1) matrices over Z[t,t ) 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 problem 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 essential 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 representation of B . The trace is invariant on Markov's equivalence n 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 of the singularities y = x n . It was realized by V. Arnold and E. Brieskorn that the fundamental groups of the complement of discriminats of semi-universal deformations of other singularities 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 corresponding 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 systems. 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 functions 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 singularities1 ). Presentations for the corresponding fundamental groups were found by Van der Lek as extended Artin groups2 ). Results on the relationship between the cohomology of pure braid groups and Weyl groups was extended 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 singularities, 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 complement 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 singularities (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 generalized 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 s 2 . 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

relat~on . to operator a 1 ge b ra appears ~n. l) wh ere Jones o b - 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 1 [M:N]. Jones showed in ) that for II1 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 M = N, M = M, and Mi =End (M.), M. being 0 1 M. 1 ~ ~ ~- a right M. -module. Note that Mi-ls Mi. Inside Mi is the ~- 1 orthogonal projection onto Mi-l which we call e .. If one makes -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 re 1 at~ng. entropy to t h e ~n . d ex ques t'~on l) . 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 interesting 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

subject of new work by Holmes an d W1'11' 1ams l) . 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 complex 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 d now as the root space. If ~ is the diagonal in E , and ~ d the discriminant in pd, then n 1 (E -6) is the pure braid group and ~ 1 (Pd-~) the braid group. The elementary synmetric functions yield a natural map f: Ed ~ Pd. Root finding algorithms yield local sections for f. 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

1) within £ The cohomology of the braid group plays an important 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, coherent 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 interdisciplinary 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

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