PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1998 | 42 | 1 | 421-446
Tytuł artykułu

Homology of braid groups and their generalizations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In the paper we give a survey of (co)homologies of braid groups and groups connected with them. Among these groups are pure braid groups and generalized braid groups. We present explicit formulations of some theorems of V. I. Arnold, E. Brieskorn, D. B. Fuks, F. Cohen, V. V. Goryunov and others. The ideas of some proofs are outlined. As an application of (co)homologies of braid groups we study the Thom spectra of these groups.
Rocznik
Tom
42
Numer
1
Strony
421-446
Opis fizyczny
Daty
wydano
1998
Twórcy
  • Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia, cr
Bibliografia
  • [Ad] J. F. Adams, Stable homotopy and generalised homology, The University of Chicago Press, Chicago and London, 1974.
  • [Arn1] V. I. Arnold, The cohomology ring of colored braids, Mat. Zametki 5 No 2 (1969), 227-231 (Russian), English transl. in Trans. Moscow Math. Soc. 21 (1970), 30-52.
  • [Arn2] V. I. Arnold, On some topological invariants of algebraic functions, Trudy Moskov. Mat. Obshch. 21 (1970), 27-46 (in Russian), English transl. in Trans. Moscow Math. Soc. 21 (1970), 30-52.
  • [Art1] E. Artin, Theorie der Zopfe, Abh. math. semin. Univ. Hamburg 4 (1925), 47-72.
  • [Art2] E. Artin, Theory of braids, Ann. of Math. 48, No 1 (1947) 101-126.
  • [Bi] J. Birman, Braids, links, and mapping class groups, Ann. Math. Stud., No 82, 1974.
  • [Bo] N. Bourbaki, Groupes et algèbres de Lie. Chap. 4, 5, 6., Hermann, Paris, 1968.
  • [BCKQRS] A. Bousfield, E. Curtis, D. Kan, D. Quillen, D. Rector, J. Schlesinger, The mod p lower central series and the Adams spectral sequence, Topology 5 (1966), 331-342.
  • [Bri] E. Brieskorn, Sur les groupes de tresses, Sém. Bourbaki, n°401, novembre 1971 (Lecture Notes in Math., No 317, 1973, 21-44).
  • [BG] E. Brown, S. Gitler, A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology 12 (1973), 283-295.
  • [BP] E. Brown, F. Peterson, The stable decomposition of $Ω^2 S^{r+2}$, Trans. Amer. Math. Soc. 243 (1978), 287-298.
  • [Bro] K. S. Brown, Cohomology of groups, Springer, N. Y. a. o., 1982.
  • [Bu] S. Bullett, Permutations and braids in cobordism theory, Proc. London Math. Soc. 38, Part 3 (1979) 517-531.
  • [Ch] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49, No 3 (1948), 654-658.
  • [CF1] F. Cohen, Cohomology of braid spaces, Bull. Amer. Math. Soc. 79 No 4 (1973), 763-766.
  • [CF2] F. Cohen, Homology of $Ω^{n+1}Σ^{n+1}X$ and $C_{n+1}X,n > 0$, Bull. Amer. Math. Soc. 79 No 6 (1973), 1236-1241.
  • [CF3] F. Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978), 99-110.
  • [CF4] F. Cohen, Artin's braid groups, classical homotopy theory, and other curiosities, Braids (Contemp. Math. 78, 1988), 167-206.
  • [CLM] F. Cohen, T. Lada, J. P. May, The homology of iterated loop spaces, (Lecture Notes in Math.; No 533), Springer-Verlag, Berlin a. o., 1976.
  • [CT] F. Cohen and L. Taylor, On the representation theory associated to the cohomology of configuration spaces, Algebraic Topology. Oaxtepec 1991, Contemp. Math. 146 (1993), 91-109.
  • [CR] R. Cohen, The geometry of $Ω^2 S^3$ and braid orientations, Invent. Math. 54 (1979), 53-67.
  • [D] P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273-302.
  • [DL] E. Dyer and R. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 No 1 (1962), 35-88.
  • [FaN] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 Fasc. I (1962), 111-118.
  • [FoN] R. Fox and L. Neuwirth, The braid groups, Math. Scand. 10 Fasc. I (1962), 119-126.
  • [FK] R. Fricke, F. Klein, Vorlesungen über die Theorie der automorphen Functionen. Bd. I. Gruppentheoretischen Grundlagen, Teubner, Leipzig, 1897 (Johnson Repr. Corp., N. Y., 1965, 634 p.).
  • [F1] D. B. Fuks, Cohomology of the braid group mod 2, Funktsional. Anal. i Prilozh. 4, No 2 (1970), 62-75 (in Russian), English transl. in Functional Anal. Appl. 4 (1970), 143-151.
  • [F2] D. B. Fuks, Quillenization and bordisms, Funktsional. Anal. i Prilozh. 8, No 1 (1974), 36-42 (in Russian), English transl. in Functional Anal. Appl. 8 (1974), 31-36.
  • [G1] V. V. Goryunov, Cohomology of the braid groups of the series C and D and some stratifications, Funktsional Anal. i Prilozh. 12, No 2 (1978), 76-77 (in Russian), English transl. in Functional Anal. Appl. 12 (1978), 139-140.
  • [G2] V. V. Goryunov, Cohomology of the braid groups of the series C and D, Trudy Moskov. Mat. Obshch. 42 (1981), 234-242 (in Russian), English transl. in Trans. Moscow Math. Soc. 1982, no 2.
  • [H] A. Hurwitz, Über Riemannsche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1-61.
  • [La] S. Lambropoulou, Solid torus links and Hecke algebras of type B, in the Proceedings of the Conference on Quantum Topology, ed. D. N. Yetter, World Scientific Press, 1993, 225-245.
  • [Li] V. Ya. Lin, Artinian braids and groups and spaces connected with them, Itogi Nauki i Tekhniki (Algebra, Topologiya, Geometriya) 17 (1979), 159-227 (in Russian). English transl. in J. Soviet Math. 18 (1982) 736-788.
  • [Mah1] M. Mahowald, A new family in $π_{*}^{s}$, Topology 16 (1977), 249-254.
  • [Mah2] M. Mahowald, Ring spectra which are Thom complexes, Duke Math. J. 46, No 3 (1977), 249-259.
  • [May] J. P. May, The Geometry of iterated loop spaces, (Lecture Notes in Math.; No 271) Springer-Verlag, Berlin a. o., 1972.
  • [O] E. Ossa, On the cohomology of configuration spaces, Algebraic Topology: New Trends in Localization and Periodicity (Barcelona Conference on Algebraic Topology, 1994) Birkhäuser Verlag, Basel a. o., 1996, 353-361.
  • [Sa] B. Sanderson, The Geometry of Mahowald Orientations, in: Algebraic Topology. Aarhus, 1978 (Lecture Notes in Math., No 533) Springer-Verlag, Berlin a. o. (1979), 152-174.
  • [Se] G. Segal, Configuration spaces and iterated loop spaces, Invent. Math. 21 (1973), 213-221.
  • [St] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, 1968.
  • [Sw] R. M. Switzer, Algebraic Topology - Homotopy and Homology, Springer-Verlag Berlin a. o., 1975.
  • [Vai] F. V. Vainshtein, Cohomology of the braid groups, Funktsional. Anal. i Prilozh. 12, No 2 (1978), 72-73 (in Russian), English transl. in Functional Anal. Appl. 4 (1970), 143-151.
  • [Vas] V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, (Translations of mathematical monographs; v. 98), AMS, Providence, 1992.
  • [Ve] V. V. Vershinin, Thom spectra of generalized braid groups, Preprint No 95/02-2, Université de Nantes, 1995.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv42i1p421bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.