Homology of braid groups and their generalizations
Vershinin, Vladimir
Banach Center Publications, Tome 43 (1998), p. 421-446 / Harvested from The Polish Digital Mathematics Library

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.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:208821
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Vershinin, Vladimir. Homology of braid groups and their generalizations. Banach Center Publications, Tome 43 (1998) pp. 421-446. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv42i1p421bwm/

[000] [Ad] J. F. Adams, Stable homotopy and generalised homology, The University of Chicago Press, Chicago and London, 1974. | Zbl 0309.55016

[001] [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.

[002] [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.

[003] [Art1] E. Artin, Theorie der Zopfe, Abh. math. semin. Univ. Hamburg 4 (1925), 47-72. | Zbl 51.0450.01

[004] [Art2] E. Artin, Theory of braids, Ann. of Math. 48, No 1 (1947) 101-126. | Zbl 0030.17703

[005] [Bi] J. Birman, Braids, links, and mapping class groups, Ann. Math. Stud., No 82, 1974.

[006] [Bo] N. Bourbaki, Groupes et algèbres de Lie. Chap. 4, 5, 6., Hermann, Paris, 1968.

[007] [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. | Zbl 0158.20502

[008] [Bri] E. Brieskorn, Sur les groupes de tresses, Sém. Bourbaki, n°401, novembre 1971 (Lecture Notes in Math., No 317, 1973, 21-44).

[009] [BG] E. Brown, S. Gitler, A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology 12 (1973), 283-295. | Zbl 0266.55012

[010] [BP] E. Brown, F. Peterson, The stable decomposition of Ω2Sr+2, Trans. Amer. Math. Soc. 243 (1978), 287-298. | Zbl 0404.55003

[011] [Bro] K. S. Brown, Cohomology of groups, Springer, N. Y. a. o., 1982.

[012] [Bu] S. Bullett, Permutations and braids in cobordism theory, Proc. London Math. Soc. 38, Part 3 (1979) 517-531. | Zbl 0405.55002

[013] [Ch] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49, No 3 (1948), 654-658. | Zbl 0033.01002

[014] [CF1] F. Cohen, Cohomology of braid spaces, Bull. Amer. Math. Soc. 79 No 4 (1973), 763-766. | Zbl 0272.55012

[015] [CF2] F. Cohen, Homology of Ωn+1Σn+1X and Cn+1X,n>0, Bull. Amer. Math. Soc. 79 No 6 (1973), 1236-1241.

[016] [CF3] F. Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978), 99-110. | Zbl 0377.55008

[017] [CF4] F. Cohen, Artin's braid groups, classical homotopy theory, and other curiosities, Braids (Contemp. Math. 78, 1988), 167-206.

[018] [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. | Zbl 0334.55009

[019] [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.

[020] [CR] R. Cohen, The geometry of Ω2S3 and braid orientations, Invent. Math. 54 (1979), 53-67.

[021] [D] P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273-302. | Zbl 0238.20034

[022] [DL] E. Dyer and R. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 No 1 (1962), 35-88. | Zbl 0119.18206

[023] [FaN] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 Fasc. I (1962), 111-118. | Zbl 0136.44104

[024] [FoN] R. Fox and L. Neuwirth, The braid groups, Math. Scand. 10 Fasc. I (1962), 119-126. | Zbl 0117.41101

[025] [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.).

[026] [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.

[027] [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.

[028] [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.

[029] [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.

[030] [H] A. Hurwitz, Über Riemannsche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1-61.

[031] [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. | Zbl 0884.57004

[032] [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.

[033] [Mah1] M. Mahowald, A new family in π*s, Topology 16 (1977), 249-254.

[034] [Mah2] M. Mahowald, Ring spectra which are Thom complexes, Duke Math. J. 46, No 3 (1977), 249-259.

[035] [May] J. P. May, The Geometry of iterated loop spaces, (Lecture Notes in Math.; No 271) Springer-Verlag, Berlin a. o., 1972. | Zbl 0244.55009

[036] [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.

[037] [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.

[038] [Se] G. Segal, Configuration spaces and iterated loop spaces, Invent. Math. 21 (1973), 213-221. | Zbl 0267.55020

[039] [St] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, 1968.

[040] [Sw] R. M. Switzer, Algebraic Topology - Homotopy and Homology, Springer-Verlag Berlin a. o., 1975. | Zbl 0305.55001

[041] [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.

[042] [Vas] V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, (Translations of mathematical monographs; v. 98), AMS, Providence, 1992.

[043] [Ve] V. V. Vershinin, Thom spectra of generalized braid groups, Preprint No 95/02-2, Université de Nantes, 1995.