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Artin braids and the groups and spaces connected with them

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Abstract

Papers on braid theory and some of its generalization and applications, reviewed in Referativnyi Zhurnal “Matematika” during 1953–1977, as well as individual papers on an earlier period, are surveyed.

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Literature cited

  1. V. I. Arnol'd, “A remark on the branching of hyperelliptic integrals as functions of the parameters,” Funkts. Analiz Ego Prilozhen.,2, No. 3, 1–3 (1968).

    Google Scholar 

  2. V. I. Arnol'd, “On the braids of algebraic functions and the cohomologies of swallowtails,” Usp. Mat. Nauk,23, No. 4, 247–248 (1968).

    Google Scholar 

  3. V. I. Arnol'd, “Cohomology ring of the group of dyed braids,” Mat. Zametki,5, No. 2, 227–231 (1969).

    Google Scholar 

  4. V. I. Arnol'd. “On cohomology classes of algebraic functions preserved under Tschirnhausen transformations,” Funkts. Analiz Ego Prilozhen.,4, No. 1, 84–85 (1970).

    Google Scholar 

  5. V. I. Arnol'd, “On certain topological invariants of algebraic functions,” Tr. Mosk. Obshch.,21, 27–46 (1970).

    Google Scholar 

  6. V. I. Arnol'd, “Topological invariants of algebraic functions. II,” Funkts. Analiz Ego Prilozhen.,4, No. 2, 1–9 (1970).

    Google Scholar 

  7. V. I. Arnol'd, “Normal forms of functions close to degenerate critical points, Weyl groups Ak. Dk. Ek. and Lagrange singularities,” Funkts. Analiz Ego Prilozhen.,6, No. 4, 3–25 (1972).

    Google Scholar 

  8. V. I. Arnol'd, “Critical points of functions and classification of caustics,” Usp. Mat. Nauk,29, No. 3, 243–244 (1974).

    Google Scholar 

  9. V. I. Arnol'd, “Critical points of smooth functions and their normal forms,” Usp. Mat. Nauk,30, No. 5, 3–65 (1975).

    Google Scholar 

  10. V. I. Arnol'd, “Critical points of functions on a manifold with boundary, simple Lie groups Bk. Ck. F4. and singularities of evolutes,” Usp. Mat. Nauk,33, No. 5, 91–105 (1978).

    Google Scholar 

  11. N. Bourbaki, Éléments de Mathématique. Fase. XXXIV. Groupes et Algèbres de Lie. Chapitre IV: Groupes de Coxeter et Systèmes de Tits. Chapitre V: Groupes Engendrés par des Réflexions. Chapitre VI: Systèmes de Racines. Hermann et Cie., Paris (1968).

    Google Scholar 

  12. N. M. Vainberg, “On the free equivalence of closed braids,” Dokl. Akad. Nauk SSSR,23, 215–216 (1939).

    Google Scholar 

  13. P. V. Vainshtein, “Cohomologies of braid groups,” Funkts. Analiz Ego Prilozhen.,12, No. 2, 72–73 (1978).

    Google Scholar 

  14. A. N. Varchenko, “On the branching of multiple integrals depending on parameters,” Funkts. Analiz Ego Prilozhen.,3, No. 4, 79–80 (1969).

    Google Scholar 

  15. É. B. Vinberg and O. V. Shvartsman, “Riemann surfaces,” in: Algebra. Topology. Geometry [in Russian], Vol. 16, Itogi Nauki i Tekh., VINITI Akad. Nauk SSSR, Moscow (1978), pp. 191–245.

    Google Scholar 

  16. O. Ya. Viro, “Links, two-sheeted branching coverings, and braids,” Mat. Sb.,87, No. 2, 216–228 (1972).

    Google Scholar 

  17. E. A. Gorin, “On the factorization of abstract entire functions,” Usp. Mat. Nauk,25, No. 4, 177–178 (1970).

    Google Scholar 

  18. E. A. Gorin, “Equations with abstract entire functions in Banach algebras,” in: Materials of Eighth Math. Interinst. Sci. Conf. Far East [in Russian], Khabarovsk (1970), pp. 9–12.

  19. E. A. Gorin, “Several examples connected with algebraic equations in algebras of functions,” Dokl. Akad. Nauk SSSR,200, No. 2, 273–276 (1971).

    Google Scholar 

  20. E. A. Gorin, “On certain algebraic equations with holomorphic coefficients,” Usp. Mat. Nauk,27, No. 3, 197–198 (1972).

    Google Scholar 

  21. E. A. Gorin, “On algebraic equations in algebras of continuous functions,” Rept. Abstr. Sixth All-Union Topol. Conf. [in Russian], Metsniereba, Tbilisi (1972), p. 43.

    Google Scholar 

  22. E. A. Gorin, “Holomorphic functions on an algebraic manifold and the reducibility of separable polynomials over certain commutative Banach Algebras,” Rept, Abstr. Seventh All-Union Topol. Conf. [in Russian], Minsk (1977), p. 55.

  23. E. A. Gorin, “Algebraic equations in commutative Banach algebras and related questions,” Zap. Nauchn. Sem. LOMI Akad. Nauk SSSR,81, 58–61 (1978).

    Google Scholar 

  24. E. A. Gorin, “On the researches of G. E. Shilov on the theory of commutative Banach algebras and their subsequent development, Usp. Mat. Nauk,33, No. 4, 169–189 (1978).

    Google Scholar 

  25. E. A. Gorin and V. Ya. Lin, “Algebraic equatons with continuous coefficients, “Material Seventh Math. and Seventh. Phys. Interinst. Sci. Conf. Far East [in Russian], Khabarovsk (1968), pp. 10–12.

  26. E. A. Gorin and V. Ya. Lin, “Braid groups and algebraic equations with continuous coefficients, “Usp. Mat. Nauk,24, No. 2, 225–226 (L969).

  27. E. A. Gorin and V. Ya. Lin, “Algebraic equations with continuous coefficients and certain questions in the algebraic theory of braids,” Mat. Sb.,78, No. 4, 579–610 (1969).

    Google Scholar 

  28. E. A. Gorin and V. Ya. Lin, “On separable polynomials over commutative Banach algebras,” Dokl. Akad. Nauk SSSR,218, No. 3, 505–508 (1974).

    Google Scholar 

  29. E. A. Gorin and C. Sánchez-Fernández, “On transcendental equations in commutative Banach algegras,” Funkt. Analiz. Ego, Prilozhen.,11, No. 1, 63–64 (1977).

    Google Scholar 

  30. V. V. Goryunov, “Cohomologies of braid groups of series C and D and certain stratifications,” Funkt. Analiz. Ego Prilozhen.,12, No. 2, 76–77 (1978).

    Google Scholar 

  31. S. Zdravkovska, “Topological classification of polynomial mappings,” Usp. Mat. Nauk,25, No. 4 179–180 (1970).

    Google Scholar 

  32. V. M. Zinde, “Commutator subgroups of Artin groups,” Usp. Mat. Nauk,30, No. 5, 207–208 (1975).

    Google Scholar 

  33. V. M. Zinde, “Certain homomorphisms of Artin groups of series Bn and Dn into groups of the same series and into symmetric groups,” Usp. Mat. Nauk,32, No. 1, 189–190 (1977).

    Google Scholar 

  34. V. M. Zinde, “Analytic properties of spaces of regular orbits of Coxeter groups of serie B and D, “Funkts. Analiz Ego Prilozhen.,11, No. 1, 69–70 (1977).

    Google Scholar 

  35. V. M. Zinde, “Homomorphisms of Artin groups of series B and D,” Moscow Univ., Moscow (1977) (Manuscript deposited in VINITI, June 6, 1977, No. 2230-77 Dep.).

    Google Scholar 

  36. V. M. Zinde, “Holomorphic mappings of spaces of regular orbits of Coxeter groups of series B and D,” Sib. Mat. Zh.,18, No. 5, 1015–1026 (1977).

    Google Scholar 

  37. Yu. V. Zyuzin, “Algebraic equations with continuous coefficients on uniform spaces,” Vestn. Mosk. Univ., No. 1, 51–53 (1972).

    Google Scholar 

  38. Yu. V. Zyuzin, “Irreducible holomorphic separable polynomials on bouquets of circular annuli,” Usp. Mat. Nauk,29, No. 5, 221–222 (1974).

    Google Scholar 

  39. Yu. V. Zyuzin, “Irreducible separable polynomials with holomorphic coefficients on a certain class of complex spaces,” Mat. Sb.,102, No. 4, 569–591 (1977).

    Google Scholar 

  40. Yu. V. Zyuzin, “Separable polynomials over functional algebras,” Author's abstract of dissertation for the academic degree of Candidate of Phys.-Math. Sciences, Mosk. Gos. Univ., Moscow (1974).

    Google Scholar 

  41. Yu. V. Zyuzin and V-Ya. Lin, “Nonbranching algebraic extensions of commutative Banach algebras,” Mat. Sb.,91, No. 3, 402–420 (1973).

    Google Scholar 

  42. Sh. I. Kaliman, “Holomorphic universal covering of the space of polynomials without multiple roots,” Funkts. Analiz Ego Prilozhen.,9, No. 1, 71 (1975).

    Google Scholar 

  43. Sh. I. Kaliman, “Holomorphic endomorphisms of the manifold of complex polynomials with, discriminant 1,” Usp. Mat. Nauk,31, No. 1, 251–252 (1976).

    Google Scholar 

  44. Sh. I. Kaliman, “Holomorphic universal covering of the space of polynomials without multiple roots,” Teoriya Funktsii, Funkt, Analiz. Ikh Prilozhen., Resp, Nauchn, Sb., No. 28, 25–35 (1977).

    Google Scholar 

  45. Sh. I. Kaliman, “Holomorphic endomorphisms of complex Weyl chambers of series Dn.” Usp. Mat. Nauk,34, No. 2, 201–202 (1979).

    Google Scholar 

  46. F. Klein, Vorlesungen über Höhere Geometrie, Dritte Auflage, Verlag von Julius Springer, Berlin (1926).

    Google Scholar 

  47. A. G. Kurosh; Theory of Groups [in Russian.], 3rd aug, ed., Nauka, Moscow (1967).

    Google Scholar 

  48. V. Ya. Lin, “Algebroid functions and holomorphic elements of homotopy groups of a complex manifold,” Dokl. Akad. Nauk SSSR,201, No. 1, 28–31 (1971).

    Google Scholar 

  49. V. Ya. Lin, “On the representations of a braid group by permutations,” Usp. Mat. Nauk.27, No. 3, 192 (1972).

    Google Scholar 

  50. V. Ya. Lin, “Algebraic functions with a universal discriminant manifold,” Funkt, Analiz Ego Prilozhen.,6, No. 1, 81–82 (1972).

    Google Scholar 

  51. V. Ya. Lin, “On the superpositions of algebraic functions,” Funkts. Analiz. Ego Prilozhen.,6, No. 3, 77–78 (1972).

    Google Scholar 

  52. V. Ya. Lin, “Representations of braids by permutations,” Usp. Mat. Nauk,29, No. 1, 173–174 (1974).

    Google Scholar 

  53. V. Ya. Lin, “On fourth-degree polynomials over the algebra of continuous functions,” Funkts. Analiz Ego Prilozhen.,8, No. 4, 89–90 (1974).

    Google Scholar 

  54. V. Ya. Lin, “Superpositions of algebraic functions,” Funkts. Analiz Ego Prilozhen.,10, No. 1, 37–45 (1976).

    Google Scholar 

  55. V. Ya. Lin, “Holomorphic mappings of certain spaces connected with algebraic functions,” Zap. Nauchn. Sem. LOMI Akad. Nauk SSSR,81, 62–65 (1978).

    Google Scholar 

  56. O. V. Lyashko, “Geometry of bifurcation diagrams,” Usp. Mat. Nauk,34, No. 3, 205–206 (1979).

    Google Scholar 

  57. W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Interscience Publishers, New York (1966).

    Google Scholar 

  58. G. S. Makanin, “The conjugacy problem in a braid group,” Dokl. Akad. Nauk SSSR,182, No. 3, 495–496 (1968).

    Google Scholar 

  59. G. S. Makanin, “On the normalizers of a braid group,” Mat. Sb.,86, No. 2, 171–179 (1971).

    Google Scholar 

  60. A. I. Mal'tsev, “On an isomorphic representation of infinite groups of matrices,” Mat. Sb.,4, 405–422 (1940).

    Google Scholar 

  61. A. A. Markov, “Über die freie Äquivalenz der geschlossener Zöpfe,” Mat. Sb.,1, 73–78 (1936).

    Google Scholar 

  62. A. A. Markov, Foundations of the Algebraic Theory of Tresses, Tr. Mat. Inst. Steklov, Vol. 16, Moscow (1945).

    Google Scholar 

  63. H. Neumann, Varieties of Groups, Springer-Verlag, Berlin-Heidelberg-New York (1967).

    Google Scholar 

  64. B. I. Plotkin, Automorphism Groups of Algebraic Systems [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  65. D. M. Smirnov, “On the theory of residually finite groups,” Ukr. Mat. Zh.,15, No. 4, 453–457 (1963).

    Google Scholar 

  66. V. B. Styshnev, “Extraction of a root in a braid group,” Izv. Akad. Nauk SSSR, Ser. Mat.,42, No. 5, 1120–1131 (1978).

    Google Scholar 

  67. D. B. Fuks, “Cohomologies of a braid group mod 2,” Funkts. Analiz Ego Prilozhen.,4, No. 2, 62–73 (1970).

    Google Scholar 

  68. D. B. Fuks, “Quillenization and Bordism,” Funkts. Analiz Ego Prilozhen.,8, No. 1, 36–42 (1974).

    Google Scholar 

  69. M. Hall, The Theory of Groups, MacMillan Co., New York (1959).

    Google Scholar 

  70. S. I. Epshtein, “Fundamental groups of spaces of collections of relatively prime polynomials,” Funkts. Analiz Ego Prilozhen.,7, No. 1, 90–91 (1973).

    Google Scholar 

  71. J. W. Alexander, “A lemma on systems of knotted curves,” Proc. Nat. Acad. Sci. U.S.A.,9, 93–95 (1923).

    Google Scholar 

  72. E. Artin, “Theorie der Zöpfe,” Abh. Math. Sem. Univ. Hamburg,4, 47–72 (1925).

    Google Scholar 

  73. E. Artin, “Theory of Braids,” Ann. Math.,48, No. 1, 101–126 (1947).

    Google Scholar 

  74. E. Artin, “Braids and permutations,” Ann. Math.,48, No. 3, 643–649 (1947).

    Google Scholar 

  75. E. Artin, “The theory of braids,” Am. Sci.,38, 112–119 (1950).

    Google Scholar 

  76. R. Baer, “Über Kurventypen auf Flächen,” J. Reine Angew. Math.,156, 231–246 (1927).

    Google Scholar 

  77. R. Baer, “Die Abbildungstypen-Gruppe der orientierbaren geschlossen Flaeche vom Geschlechte 2,” J. Reine Angew. Math.,160, 1–25 (1929).

    Google Scholar 

  78. G. Baumslag, “Automorphism groups of residually finite groups,” J. London Math. Soc.,38, No. 1, 117–118 (1963).

    Google Scholar 

  79. J. S. Birman, “Braid groups and their relationship to mapping class groups,” Doctoral Dissertation, New York Univ., New York (1968).

    Google Scholar 

  80. J. S. Birman, “On braid groups,” Commun. Pure Appl. Math.,22, No. 1, 41–72 (1969).

    Google Scholar 

  81. J. S. Birman, “Mapping class groups and their relationship to braid groups,” Commun. Pure Appl. Math.,22, No. 2, 213–238 (1969).

    Google Scholar 

  82. J. S. Birman, “Nonconjugate braids can define isotopic knots,” Commun. Pure Appl. Math.,22, No. 2, 239–242 (1969).

    Google Scholar 

  83. J. S. Birman, “Automorphisms of the fundamental group of a closed, orientable 2-manifold,” Proc. Am. Math. Soc.,21, No. 2, 351–354 (1969).

    Google Scholar 

  84. J. S. Birman, “Abelian quotients of the mapping class group of a 2-manifold,” Bull. Am. Math., Soc.,76, No. 1, 147–150 (1970).

    Google Scholar 

  85. J. S. Birman, “A normal form in the homeotopy group of a surface of genus 2, with applications to 3-manifolds,” Proc. Am. Math. Soc.,34, No. 2, 379–384 (1972).

    Google Scholar 

  86. J. S. Birman, “Plat representations for link groups,” Commun. Pure Appl. Math.,26, No. 5–6, 673–678 (1973).

    Google Scholar 

  87. J. S. Birman, “Mapping class groups of surfaces: a survey,” in: L. Greenberg (ed.), Discontinuous Groups and Riemann Surfaces, Ann. Math. Studies, No. 79, Princeton Univ. Press, Princeton, NJ (1974), pp. 57–71.

    Google Scholar 

  88. J. S. Birman, “Poincare's conjecture and the homeotopy group of a closed, orientable 2-manifold,” J. Austral. Math. Soc.,17, No. 2, 214–221 (1974).

    Google Scholar 

  89. J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. Math. Studies, No. 82, Princeton Univ. Press, Princeton, NJ (1975).

    Google Scholar 

  90. J. S. Birman, “On the stable equivalence of plat representations of knots and links,” Can. J. Math.,28, No. 2, 264–290 (1976).

    Google Scholar 

  91. J. S. Birman and D. R. J. Chillingworth, “On the homeotopy group of a nonorientable surface,” Proc. Cambridge Philos. Soc.,71, No. 3, 437–438 (1972).

    Google Scholar 

  92. J. S. Birman and H. M. Hilden, “On the mapping class groups of closed surfaces and as covering spaces,” in: L. V. Ahlfors, L. Bers, H. M. Farkas, R. C. Gunning, I. Kra, and H. E. Rauch (eds.), Advances in the Theory of Riemann Surfaces, Ann. Math. Studies, No. 66, Princeton Univ. Press, Princeton, NJ (1971), pp. 81–115.

    Google Scholar 

  93. J. S. Birman and H. M. Hilden, “Lifting and projecting homeomorphisms,” Arch. Math.,23, No. 4, 428–434 (1972).

    Google Scholar 

  94. J. S. Birman and H. M. Hilden, “Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin's braid group,” Bull. Am. Math. Soc.,78, No. 6, 1002–1004 (1972).

    Google Scholar 

  95. J. S. Birman and H. M. Hilden, “On isotopies of homeomorphisms of Riemann surfaces,” Ann. Math.,97, No. 3, 424–439 (1973).

    Google Scholar 

  96. J. S. Birman and H. M. Hilden, “The homeomorphism problem for S3” Bull. Am. Math. Soc.,79, No. 5, 1006–1009 (1973).

    Google Scholar 

  97. J. S. Birman and H. H. Hilden, “Heegaard splittings of branched coverings of S3.” Trans. Am. Math. Soc.,213, 315–352 (1975).

    Google Scholar 

  98. J. S. Birman and W. Magnus, “Discriminant and projective invariants of binary forms,” Commun. Pure Appl. Math.,23, No. 3, 269–275 (1970).

    Google Scholar 

  99. F. Bohnenblust, “The algebraic braid group,” Ann. Math.,48, No. 1, 127–136 (1947).

    Google Scholar 

  100. E. Brieskorn, “Sur les groupes de tresses [d'après V. I. Arnol'd],” in: Séminaire Bourbaki, Vol. 1971/72, Exposés 400-417, Lect. Notes Math., Vol. 317, Springer-Verlag, Berlin-Heidelberg-New York (1973), pp. 21–44.

    Google Scholar 

  101. E. Brieskorn, “Die fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe,” Invent. Math.,12, No. 1, 57–61 (1971).

    Google Scholar 

  102. E. Brieskorn and K. Saito, “Artin-Gruppen und Coxeter-Gruppen,” Invent. Math.,17, No. 4, 245–271 (1972).

    Google Scholar 

  103. W. Burau, “Über Zopfinvarianten,” Abh. Math. Sem. Univ. Hamburg,9, 117–124 (1932).

    Google Scholar 

  104. W. Burau, “Über Zopfgruppen und gleichsinning verdrillte Verkettungen,” Abh. Math. Sem. Univ. Hamburg,11, 179–186 (1935).

    Google Scholar 

  105. W. Burau, “Über Verkettungsgruppen,” Abh. Math. Sem. Univ. Hamburg,11, 171–178 (1935).

    Google Scholar 

  106. G. Burde, “Zur Theorie der Zöpfe,” Math. Ann.,151, No. 2, 101–107 (1963).

    Google Scholar 

  107. G. Burde, “Über Normalisatoren der Zopfgruppen,” Abh. Math. Sem. Univ. Hamburg,27, No. 1–2, 97–115 (1964).

    Google Scholar 

  108. G. Burde and K. Murasugi, “Links and Seifert fiber spaces,” Duke Math. J.,37, No. 1, 89–93 (1970).

    Google Scholar 

  109. W.-L. Chow, “On the algebraic braid group,” Ann. Math.,49, No. 3, 654–658 (1948).

    Google Scholar 

  110. D. B. Cohen, “The Hurwitz monodromy group,” J. Algebra,32, No. 3, 501–517 (1974).

    Google Scholar 

  111. D. I. A. Cohen, “On representations of the braid group,” J. Algebra,7, No. 2, 145–151 (1967).

    Google Scholar 

  112. F. R. Cohen, “Braid orientations and bundles with flat connections,” Invent. Math.,46, No. 2, 99–110 (1978).

    Google Scholar 

  113. F. R. Cohen, T. J. Lada, and J. P. May, The Homology of Iterated Loop Spaces, Lect. Notes Math., Vol. 533, Springer-Verlag, Berlin-Heidelberg-New York (1976).

    Google Scholar 

  114. T. M. Cowan, “The theory of braids and the analysis of impossible figures,” J. Math. Psychol.,11, No. 3, 190–212 (1974).

    Google Scholar 

  115. D. M. Dahm, “A generalization of braid theory,” Doctoral Dissertation, Princeton Univ., Princeton, NJ (1962).

    Google Scholar 

  116. M. Dehn, “Die Gruppe der Abbildungsklassen,” Acta Math.,69, 135–206 (1838).

    Google Scholar 

  117. P. Deligne, “Les immeubles des groupes de tresses généralisés,” Invent. Math.,17, No. 4, 273–302 (1972).

    Google Scholar 

  118. E. Dyer and R. K. Lashof,” Homology of iterated loop spaces,” Am. J. Math.,84, No, 1, 35–88 (1962).

    Google Scholar 

  119. M. Engber, “A strengthening of centerlessness in Teichmüller theory,” Proc. Am. Math. Soc.,60, 144–148 (1976).

    Google Scholar 

  120. E. Fadell, “Homotopy groups of configuration spaces and the string problem of Dirac,” Duke Math. J.,9, No. 2, 231–242 (1962).

    Google Scholar 

  121. E. Fadell and L. Neuwirth, “Configuration spaces,” Math. Scand.,10, No. 1, 111–118 (1962).

    Google Scholar 

  122. E. Fadell and J. Van Buskirk, “On the braid groups of E2 and S2.” Bull. Am. Math. Soc.,67, No. 2, 211–213 (1961).

    Google Scholar 

  123. E. Fadell and J. Van Buskirk, “The braid groups of E2 and S2.” Duke Math. J.,29, No. 2, 243–257 (1962).

    Google Scholar 

  124. R. H. Fox and L. Neuwirth, “The braid groups,” Math. Scand.,10, No. 1, 119–126 (1962).

    Google Scholar 

  125. R. Fricke and E. Klein, Vorlesungen über die Theorie der automorphen Funktionen. Band I: Die Gruppentheoretischen Grundlagen. Teubner, Stuttgart (1897); Johnson Reprint Corp., New York (1965).

    Google Scholar 

  126. W. Fröhlich, “Über ein spezielles Transformationsproblem bei einer besonderen Klasse von Zöpfen,” Monatsh. Math. Phys.,44, 225–237 (1936).

    Google Scholar 

  127. F. A. Garside, “The theory of knots and associated problems,” Doctoral Dissertation, Oxford Univ., Oxford (1965).

    Google Scholar 

  128. F. A. Garside, “The braid group and other groups,” Q. J. Math.,20, No. 78, 235–254 (1969).

    Google Scholar 

  129. B. J. Gassner, “On braid groups,” Abh. Math. Sem. Univ. Hamburg,25, No. 1–2, 10–22 (1961).

    Google Scholar 

  130. R. Gillette and J. Van Buskirk, “The word problem and consequences for the braid groups and mapping class groups of the 2-sphere,” Trans. Am. Math. Soc.,131, No. 2, 277–296 (1968).

    Google Scholar 

  131. J. Gilman, “An example about normalizers in mapping class groups,” Proc. Am. Math. Soc.,69, No. 1, 115–118 (1978).

    Google Scholar 

  132. C. H. Goldberg, “An exact sequence of braid groups,” Math. Scand.,33, No. 1, 69–82 (1973).

    Google Scholar 

  133. D. L. Goldsmith, “Motions of links in the 3-sphere,” Doctoral Dissertation, Princeton Univ., Princeton, NJ (1972).

    Google Scholar 

  134. D. L. Goldsmith, “Homotopy of braids—in answer to a question of E. Artin,” in: R. F. Dickman, Jr. and P. Fletcher (eds.), Topology Conference, Lect. Notes Math., Vol. 375, Springer-Verlag, Berlin-Heidelberg-New York (1974), pp. 91–96.

    Google Scholar 

  135. D. L. Goldsmith, “Motions of links in the 3-sphere,” Bull. Am. Math. Soc.,80, No. 1, 62–66 (1974).

    Google Scholar 

  136. E. K. Grossman, “On the residual finiteness of certain mapping class groups,” J. London Math. Soc.,9, No. 1, 160–164 (1974).

    Google Scholar 

  137. E. K. Grossman, “On certain permutation representations of mapping class groups,” Math. Z.,146, No. 2, 105–112 (1976).

    Google Scholar 

  138. H. M. Hilden, “Generators for two groups related to the braid group,” Pacif. J. Math.,59, No. 2, 475–486 (1975).

    Google Scholar 

  139. A. Hurwitz, “Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten,” Math. Ann.,39, 1–61 (1891).

    Google Scholar 

  140. T. Kaneto, “On isotopy groups of holed spheres,” Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A, No. 347–365, 46–51 (1975).

    Google Scholar 

  141. T. Kaneto, “On isotopy groups of simply connected holed manifolds,” Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A, No. 347–365, 52–58 (1975).

    Google Scholar 

  142. Y. Ladegaillerie, “Groupes de tresses et problème des mots dans les groupes de tresses,” Bull. Sci. Math.,100, No. 3, 255–267 (1976).

    Google Scholar 

  143. Y. Ladegaillerie, “Decoupes et isotopies de surfaces topologiques,” Thèse Doct. Sci. Math., Univ. Sci. et Techn. Langudoc, Langudoc (1976).

    Google Scholar 

  144. H. Levinson, “Decomposable braids,” Doctoral Dissertation, New York Univ., New York (1971).

    Google Scholar 

  145. H. Levinson, “Decomposable braids and linkages,” Trans. Am. Math. Soc.,178, 111–126 (1973).

    Google Scholar 

  146. H. Levinson, “Decomposable braids as subgroups of braid groups,” Trans. Am. Math. Soc.,202, 51–55 (1975).

    Google Scholar 

  147. A. Libgober, “On the fundamental group of the space of cubic surfaces,” Math. Z.,162, No. 1, 63–67 (1978).

    Google Scholar 

  148. W. B. R. Lickorish, “Homomorphisms of nonorientable two-manifolds,” Proc. Cambridge Philos. Soc.,59, No. 2, 307–317 (1963).

    Google Scholar 

  149. W. B. R. Lickorish, “A finite set of generators for the homeotopy group of a 2-manifold,” Proc. Cambridge Philos. Soc.,60, No. 4, 769–778 (1964).

    Google Scholar 

  150. W. B. R. Lickorish, “On the homeotopy group of a 2-manifold (corrigendum),” Proc. Cambridge Philos. Soc.,62, No. 4, 679–681 (1966).

    Google Scholar 

  151. S. Lipschutz, “On a finite matrix representation of the braid group,” Arch. Math.,12, No. 1, 7–12 (1961).

    Google Scholar 

  152. S. Lipschutz, “Ntate on a paper by Shepperd on the braid group,” Proc. Am. Math. Soc.,14, No. 2, 225–227 (1963).

    Google Scholar 

  153. E. Looijenga, “The complement of the bifurcation variety of a simple singularity,” Invent. Math.,23, No. 2, 105–116 (1974).

    Google Scholar 

  154. C. Maclachlan, “On a conjecture of Magnus on the Hurwitz monodromy group,” Math. Z.,132, No. 1, 45–50 (1973).

    Google Scholar 

  155. C. Maclachlan, “Note on the Hurwitz-Nielsen realization problem,” Proc. Am. Math. Soc.,64, No. 1, 87–90 (1977).

    Google Scholar 

  156. C. Maclachlan, “On representations of Artin's braid group,” Mich. Math. J.,25, No. 2, 235–244 (1978).

    Google Scholar 

  157. C. Maclachlan and W. J. Harvey, “On mapping-class groups and Teichmüller spaces,” Proc. London Math. Soc.,30, No. 4, 496–512 (1975).

    Google Scholar 

  158. W. Magnus, “Über Automorphismen von Fundamentalgruppen berandeter Flachen,” Math. Ann.,109, No. 5, 617–646 (1934).

    Google Scholar 

  159. W. Magnus,” Residually finite groups,” Bull. Am. Math. Soc.,75, No. 2, 305–316 (1969).

    Google Scholar 

  160. W. Magnus, “Braids and Riemann surfaces,” Commun. Pure Appl. Math.,25, No. 2, 151–161 (1972).

    Google Scholar 

  161. W. Magnus, “Braid groups: a survey,” in: M. F. Newman (ed.), The Theory of Groups, Lect. Notes Math., Vol. 372, Springer-Verlag, Berlin-Heidelberg-New York (1974), pp. 463–487.

    Google Scholar 

  162. W. Magnus, “Two generator subgroups of PSL(2,C),” Nachr. Akad. Wiss, Göttingen, Math. Phys. Kl., II, No. 7, 1–14 (1975).

    Google Scholar 

  163. W. Magnus and A. Peluso, On knot groups,” Commun. Pure Appl. Math.,20, No, 4, 749–770 (1967).

    Google Scholar 

  164. W. Magnus and A. Peluso, “On a theorem of V. I. Arnol'd,” Commun. Pure Appl. Math.,22, No. 5, 683–692 (1969).

    Google Scholar 

  165. W. Mangier, “Die Klassen von topologischen Abbildungen einer geschlossenen Flache auf sich,” Math. Z.,44, No. 4, 541–554 (1939).

    Google Scholar 

  166. G. S. McCarthy, Jr., “Homeotopy groups,” Trans. Am. Math. Soc.,106, No. 2, 293–304 (1963).

    Google Scholar 

  167. J. W. Milnor, “Link groups,” Ann. Math.,59, No. 2, 177–195 (1954).

    Google Scholar 

  168. J. W. Milnor, “Isotopy of links,” in: R. H. Fox, D. C. Spencer, and A. W. Tucker (eds.), Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz, Princeton Univ. Press, Princeton, N. J. (1957), pp. 280–306.

    Google Scholar 

  169. K. Murasugi, On Closed 3-Braids, Mem. Am. Math. Soc., No. 151, Am. Math. Soc., Providence, RI (1974).

    Google Scholar 

  170. K. Murasagi, “On the divisibility of knot groups,” Pacif. J. Math.,52, No. 2, 491–503 (1974).

    Google Scholar 

  171. K. Murasugi, “Seifert fiber spaces and braid groups,” Preprint, Toronto (1979).

  172. K. Murasugi and R. S. D. Thomas, “Isotppic closed nonconjugate braids,” Proc. Am. Math. Soc.,33, No. 1, 137–139 (1972).

    Google Scholar 

  173. M. H. A. Newman, “On a string problem of Dirac,” J. London Math. Soc.,17, No. 3, 173–177 (1942).

    Google Scholar 

  174. J. Nielsen,” Über topologische Abbildungen geschlossener Flächen,” Abh. Math. Sem. Univ. Hamburg,3, 246–260 (1924).

    Google Scholar 

  175. J. Nielsen, “Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, I–III,” Acta Math.,50, 189–358 (1927);53, 1–76 (1929);58, 87–167 (1931).

    Google Scholar 

  176. J. Powell, “Two theorems on the mapping class group of a surface,” Proc. Am. Math. Soc.,68, No. 3, 347–350 (1978).

    Google Scholar 

  177. L. V. Quintas, “Solved and unsolved problems in the computation of homeotopy groups of 2-manifolds,” Trans. N. Y. Acad. Sci.,30, No. 7, 919–938 (1968).

    Google Scholar 

  178. G. P. Scott, “The space of homeomorphisms of a 2-manifold,” Topology,9, No. 1, 97–109 (1970).

    Google Scholar 

  179. G. P. Scott, “Braid groups and the group of homeomorphisms of a surface,” Proc. Cambridge Philos. Soc.,68, No. 3, 605–617 (1970).

    Google Scholar 

  180. G. B. Segal, “Configuration-spaces and iterated loop-spaces,” Invent. Math.,21, No. 3, 213–221 (1973).

    Google Scholar 

  181. J. A. H. Shepperd, “Braids which can be plaited with their threads tied together at each end,” Proc. R. Soc.,A265, No. 1321, 229–244 (1962).

    Google Scholar 

  182. N. Smythe, “Isotopy invariants of links,” Doctoral Dissertation, Princeton Univ., Princeton, NJ (1965).

    Google Scholar 

  183. D. J. Sprows, “Homeotopy groups of compact 2-manifolds,” Fund, Math.,90, No. 1, 99–103 (1975).

    Google Scholar 

  184. R. S. D. Thomas, “An algorithm for combing braids,” Proc. 2nd Louisiana Conf, Combinatorics, Graph Theory, and Computing, Baton Rouge, LA (1971), pp. 517–532.

  185. R. S. D. Thomas, “Partially closed braids,” Can. Math. Bull.,17, No. 1, 99–107 (1974).

    Google Scholar 

  186. R. S. D. Thomas, “The structure of the fundamental braids,” Q. J. Math.,26, No. 103, 283–288 (1975).

    Google Scholar 

  187. R. S. D. Thomas and B. T. Paley, “Garside's braid-conjugacy solution implemented,” Util. Math.,6, 321–335 (1974).

    Google Scholar 

  188. J. Tits, “Le problème des mots dans les groupes de Coxeter,” Symposia Mathematica, Vol. I, Academic Press Inc. (London) Ltd., London-New York (1969), pp. 175–185.

    Google Scholar 

  189. J. Van Buskirk, “Braid groups of compact 2-manifolds with elements of finite order,” Trans. Am. Math. Soc.,122, No. 1, 81–97 (1966).

    Google Scholar 

  190. H. Zieschang, “On the homeotopy groups of surfaces,” Math. Ann.,206, No. 1, 1–21 (1973).

    Google Scholar 

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Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 17, pp. 159–227, 1979.

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Lin, V.Y. Artin braids and the groups and spaces connected with them. J Math Sci 18, 736–788 (1982). https://doi.org/10.1007/BF01091963

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