No Access

AUTOMORPHISMS OF BRAID GROUPS ON S², T², P² AND THE KLEIN BOTTLE K

PING ZHANG

Department of Mathematics, Eastern Mediterranean University, Gazimagusa, North Cyprus, via Mersin 10, Turkey

https://doi.org/10.1142/S0218216508005951Cited by:3 (Source: Crossref)

Abstract

It is shown that for the braid group B_n(M) on a closed surface M of nonnegative Euler characteristic, Out(B_n(M)) is isomorphic to a group extension of the group of central automorphisms of B_n(M) by the extended mapping class group of M, with an explicit and complete description of Aut(B_n(S²)), Aut(B_n(P²)), Out(B_n(S²)) and Out(B_n(P²)).

Keywords:

AMSC: 57N05, 20F28, 20F36

References

E. Artin, Ann. Math. 48, 101 (1947). Crossref, Web of Science, Google Scholar
E. Artin, Ann. Math. 48, 643 (1947). Crossref, Web of Science, Google Scholar
T. E. Brendle and D. Margalit, Geom. Topol. 8, 1361 (2004), DOI: 10.2140/gt.2004.8.1361. Crossref, Web of Science, Google Scholar
J. S. Birman, Commun. Pure Appl. Math. 22, 213 (1969), DOI: 10.1002/cpa.3160220206. Crossref, Web of Science, Google Scholar
J. S. Birman , Braids, Links, and Mapping Class Groups ( Princeton University Press , Princeton, New Jersey , 1974 ) . Google Scholar
J. L. Dyer and E. K. Grossman, Amer. J. Math. 103, 1151 (1981), DOI: 10.2307/2374228. Crossref, Web of Science, Google Scholar
E. Fadell and J. Van Buskirk, Duke Math. J. 29, 243 (1962), DOI: 10.1215/S0012-7094-62-02925-3. Crossref, Web of Science, Google Scholar
B. Farb and N. V. Ivanov, Math. Res. Lett. 12, 293 (2005). Crossref, Web of Science, Google Scholar
R. Fox and L. Neuwirth, Math. Scand. 10, 119 (1962). Crossref, Web of Science, Google Scholar
M.-E. Hamstrom, Illinois J. Math. 10, 563 (1966). Crossref, Web of Science, Google Scholar
E. Irmak, N. V. Ivanov and J. D. McCarthy, Automorphisms of surface braid groups, preprint, 3 Jun 2003 , arXiv:math.GT/0306069 v1 . Google Scholar
N. V. Ivanov, Algebraic properties of the Teichmüller modular group (Russian), Dokl. Akad. Nauk SSSR 275 (1984) 786–789; English translation in Soviet Math. Dokl. 29 (1984) 288–291 . Google Scholar
N. V. Ivanov, Topology and Geometry–Rohlin Seminar, Lecture Notes in Mathematics 1346 (Springer, Berlin, 1988) pp. 199–270. Crossref, Google Scholar
N. V. Ivanov, Permutation representations of braid groups of surfaces, Mat. Sb. 181 (1990) 1464–1474 (in Russian); English translation in Math. USSR-Sb. 71 (1992) 309–318 . Google Scholar
N. V. Ivanov, Int. Math. Res. Not. 651 (1997), DOI: 10.1155/S1073792897000433. Google Scholar
N. V. Ivanov and J. D. McCarthy, Invent. Math. 135, 425 (1999), DOI: 10.1007/s002220050292. Crossref, Web of Science, Google Scholar
M. Korkmaz, Topology Appl. 95, 85 (1999), DOI: 10.1016/S0166-8641(97)00278-2. Crossref, Web of Science, Google Scholar
W. B. R. Lickorish, Proc. Cambridge Phil. Soc. 59, 307 (1963). Crossref, Web of Science, Google Scholar
J. D. McCarthy, Invent. Math. 84, 49 (1986), DOI: 10.1007/BF01388732. Crossref, Web of Science, Google Scholar
J. D. McCarthy and W. R. Vautaw, Automorphisms of Torelli groups, preprint, 14 Nov 2003 , arXiv:math.GT/0311250 v1 . Google Scholar
J. Nielsen, Danske Selsk. Mat.-Fis. 1 (1937). Google Scholar
L. Paris and D. Rolfsen, Ann. Inst. Fourier (Grenoble) 49, 417 (1999). Crossref, Web of Science, Google Scholar
J. S. Rose, Proc. London Math. Soc. 31, 167 (1975). Web of Science, Google Scholar
J. S. Rose , A Course on Group Theory ( Cambridge University Press , Cambridge , 1978 ) . Google Scholar
G. P. Scott, Proc. Cambridge Phil. Soc. 68, 605 (1970). Crossref, Web of Science, Google Scholar
J. Van Buskirk, Trans. Amer. Math. Soc. 122, 81 (1966), DOI: 10.2307/1994502. Crossref, Web of Science, Google Scholar
P. Zhang, J. Knot Theory Ramifications 15, 1231 (2006), DOI: 10.1142/S0218216506005044. Link, Web of Science, Google Scholar