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Loop spaces of configuration spaces,braid-like groups, and knots

  • F. R. Cohen*
  • S. Gitler*
Conference paper
Part of the Progress in Mathematics book series (PM, volume 196)

Abstract

The purpose of this note is to describe some relationships between the following topics: (1) higher dimensional variations of braids, (2) loop space homology, (3) Hopf algebras given by loop space homology, (4) natural groups attached to connected Hopf algebras, (5) analogues of Artin’s (pure) braid group, (6) Alexander’s construction of knots arising from loop spaces, and (7) Vassiliev’s invariants of braids.

Keywords

Hopf Algebra Configuration Space Braid Group Homotopy Group Loop Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. [1]
    J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Science USA 9 (1923), 93–95.CrossRefGoogle Scholar
  2. [2]
    E. Artin, The theory of braids, Ann. of Math. 48 (1947), 101–126.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    D. Bar Natan, On the Vassiliev invariants, Topology 34 (1995), 423–472.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    J. Birman, Braids, Links, and Mapping Class Groups, Ann. of Math. Studies 66 (1971).Google Scholar
  5. [5]
    A. K. Bousfield, and D. M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Math. 304, Springer-Verlag (1972).zbMATHCrossRefGoogle Scholar
  6. [6]
    F. R. Cohen, The homology of Cn+1-spaces, Lecture Notes in Math. 533, Springer-Verlag (1976), 207–351.Google Scholar
  7. [7]
    F. R. Cohen, S. Gitler, On loop spaces of configuration spaces, preprint.Google Scholar
  8. [8]
    F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. 110 (1979), 549–565.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    F. R. Cohen, T. Sato, On the group of homotopy groups, student topology seminar, Univ. of Rochester, Spring 1998, preprint.Google Scholar
  10. [10]
    F. R. Cohen, L. R. Taylor, On the representation theory associated to the cohomology of configuration spaces, Contemp. Math. 146 (1993), 91–109.MathSciNetGoogle Scholar
  11. [11]
    E. Fadell and S. Husseini, Geometry and Topology of Configuration Spaces, in preparation.Google Scholar
  12. [12]
    E. Fadell and S. Husseini, The space of loops on configuration spaces and the Majer-Terracini index, Topological Methods in Nonlinear Analysis, Journal of the Julius Schauder Center (1996).Google Scholar
  13. [13]
    E. Fadell, and L. Neuwirth, Configuration Spaces, Math. Scand. 10 (1962), 111–118.MathSciNetzbMATHGoogle Scholar
  14. [14]
    M. Falk, and R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), 77–88.MathSciNetzbMATHGoogle Scholar
  15. [15]
    T. G. Goodwillie, Cyclic homology, derivations, and Hochschild homology, Topology 24 (1985), 187–215.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    R. Hain, Infinitesimal presentations of the Torelli group, J. Amer. Math. Soc. 10 (1997), 597–691.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    I. M. James, Stiefel Manifolds, London Mathematical Society Lecture Note Series 78 (1965).Google Scholar
  18. [18]
    T. Kohno, Linear represenations of braid groups and classical Yang—Baxter equations, Contemp. Math. 78 (1988), 339–363.MathSciNetCrossRefGoogle Scholar
  19. [19]
    T. Kohno, Vassiliev invariants and de Rham complex on the space of knots, Contemp. Math. 179 (1994), 123–138.MathSciNetCrossRefGoogle Scholar
  20. [20]
    M. Kontsevich, Vassiliev’s knot invariants, Advances in Soviet Mathematics 16 (1993), 137–150.MathSciNetGoogle Scholar
  21. [21]
    J.-L. Loday, Cyclic Homology, Grundlehren der mathematischen Wissenschaften 301 (1992), Springer-Verlag, Berlin.Google Scholar
  22. [22]
    W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory (1976), Dover.Google Scholar
  23. [23]
    J. W. Milnor, and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211–264.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    T. Sato, On free groups and morphisms of coalgebras, student topology seminar, Univ. of Rochester, Spring 1998, preprint.Google Scholar
  25. [25]
    P. S. Selick, 2-primary exponents for the homotopy groups of spheres, Topology 23 (1984), 97–99.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    V. Vassiliev, Cohomology of knot spaces, Theory of singularities and its applications, Amer. Math. Soc. (1992).Google Scholar
  27. [27]
    M. Xicoténcatl, Orbit configuration spaces, Thesis, Univ. of Rochester, Spring 1996.Google Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • F. R. Cohen*
    • 1
  • S. Gitler*
    • 1
  1. 1.Department of MathematicsUniversity of Rochester,RochesterUSA

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