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On homological properties of singular braids


Author: Vladimir V. Vershinin
Journal: Trans. Amer. Math. Soc. 350 (1998), 2431-2455
MSC (1991): Primary 20J05, 20F36, 20F38, 18D10, 55P35
DOI: https://doi.org/10.1090/S0002-9947-98-02048-0
MathSciNet review: 1443895
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Abstract: Homology of objects which can be considered as singular braids, or braids with crossings, is studied. Such braids were introduced in connection with Vassiliev's theory of invariants of knots and links. The corresponding algebraic objects are the braid-permutation group $BP_{n}$ of R. Fenn, R. Rimányi and C. Rourke and the Baez-Birman monoid $SB_{n}$ which embeds into the singular braid group $SG_{n}$. The following splittings are proved for the plus-constructions of the classifying spaces of the infinite braid-permutation group and the singular braid group

\begin{equation*}\mathbb{Z}\times BBP_{\infty }^{+}\simeq \Omega ^{\infty }S^{\infty }\times S^{1} \times Y, \end{equation*}

\begin{equation*}\mathbb{Z}\times BSG_{\infty }^{+}\simeq S^{1}\times \Omega ^{2} S^{2}\times W, \end{equation*}

where $Y$ is an infinite loop space and $W$ is a double loop space.


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  • [Ad] J. F. Adams, Infinite loop spaces, Ann. Math. Stud., No 90, Princeton Univ. Press and Univ. of Tokyo Press, 1978. MR 80d:55001
  • [Arn1] V. I. Arnold, On some topological invariants of algebraic functions, Trudy Moskov. Mat. Obshch. 21 (1970), 27-46 (Russian), English transl. in Trans. Moscow Math. Soc. 21 (1970), 30-52. MR 43:225
  • [Arn2] V. I. Arnold, Topological invariants of algebraic functions II, Funk. Anal. i Prilozhen. 4, No 2 (1970), 1-9 (Russian), English transl. in Functional Anal. Appl. 4 (1970), 91-98. MR 43:1991
  • [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 (1947), 101-126. MR 8:367a
  • [Bae] John C. Baez, Link invariants of finite type and perturbation theory, Letters in Math. Physics 26 (1992), 43-51. MR 93k:57006
  • [Bar] M. G. Barratt, A free group functor for stable homotopy, Algebraic Topology (Proc. of Symposia in Pure Math. No. 22) AMS, Providence, 1971, 31-35. MR 48:3043
  • [Bat] M. Batanin, Private communication.
  • [Bi] Joan S. Birman, New points of view in knot theory, Bull. of the Amer. Math. Soc. 28 (1993), 253-387. MR 94b:57007
  • [CF1] F. Cohen, Cohomology of braid spaces, Bull. of the Amer. Math. Soc. 79 (1973), 763-766. MR 47:9607
  • [CF2] F. Cohen, Homology of $\Omega ^{n+1}\Sigma ^{n+1}X$ and $C_{n+1}X, \ n>0$, Bull. of the Amer. Math. Soc. 79 (1973), 1236-1241. MR 49:3939
  • [CF3] F. Cohen, Braid orientations and bundles with flat connections, Inventiones Math. 46 (1978), 99-110. MR 80b:57033
  • [CLM] F. Cohen, T. Lada and J. P. May, The homology of iterated loop spaces, (Lecture Notes in Math.; No 533), Springer-Verlag, Berlin a. o., 1976. MR 55:9096
  • [FKR] R. Fenn, E. Keyman and C. Rourke, The singular braid monoid embeds in a group, Preprint, 1996.
  • [FRR1] R. Fenn, R. Rimányi and C. Rourke, Some remarks on the braid-permutation group, Topics in Knot Theory. Kluwer Academic Publishers, 1993, 57-68. MR 95g:57022
  • [FRR2] R. Fenn, R. Rimányi and C. Rourke, The Braid-Permutation Group, Topology 36 (1997), 123-135. MR 97g:20041
  • [Fi] Z. Fiedorowicz, Operads and iterated monoidal categories, Preprint (1995).
  • [Fuks] D. B. Fuks, Quillenization and bordisms, Funkcional. Anal. i Prilozh. 8, No 1 (1974), 36-42 (Russian), English transl. in Functional Anal. Appl. 8 (1974), 31-36. MR 49:8043
  • [H] Allen Hatcher, Homological stability for automorphism groups of free groups, Comment. Math. Helvetici 70 (1995), 39-62. MR 95k:20030
  • [J] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), 335-388. MR 89c:46092
  • [JS] A. Joyal and R. Street, Braided tensor categories, Advances in Math. 102 (1993), 20-78. MR 94m:18008
  • [ML] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics, Springer-Verlag, N.Y. a.o., 1971. MR 50:7275
  • [Mah1] M. Mahowald, A new infinite family in ${}_2\pi _{*}^{s}$, Topology 16 (1977), 249-256. MR 56:3838
  • [Mah2] M. Mahowald, Ring spectra which are Thom complexes, Duke Math. J. 46 (1979), 549-559. MR 81f:55010
  • [May] J. P. May, $E_{\infty }$ spaces, group completions, and permutative categories, New Developments in Topology (London Math. Soc. Lecture Notes Series; No 11) Cambridge, 1974, 61-93. MR 49:3915
  • [P] Stewart B. Priddy, On $\Omega ^{\infty }S^{\infty }$ and the infinite symmetric group, Algebraic Topology (Proc. of Symposia in Pure Math.; No 22) AMS, Providence, 1971, 217-220. MR 50:11226
  • [S1] G. Segal, Configuration spaces and iterated loop spaces, Inventiones Math. 21 (1973), 213-221. MR 48:9710
  • [S2] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293-312. MR 50:5782
  • [V] V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, (Translations of Mathematical Monographs, vol. 98), AMS, Providence, 1992. MR 94i:57020
  • [W] F. Waldhausen, Algebraic $K$-theory of topological spaces. II, Algebraic Topology. Aarhus 1978. (Lecture Notes in Math; No 763), Springer-Verlag, Berlin a.o., 1979, 356-394. MR 81i:18014b

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Additional Information

Vladimir V. Vershinin
Affiliation: Institute of Mathematics, Novosibirsk, 630090, Russia
Email: versh@math.nsc.ru

DOI: https://doi.org/10.1090/S0002-9947-98-02048-0
Keywords: Braid group, permutation group, homology, classifying space, loop space
Received by editor(s): August 20, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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