Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On loop spaces of configuration spaces


Authors: F. R. Cohen and S. Gitler
Journal: Trans. Amer. Math. Soc. 354 (2002), 1705-1748
MSC (2000): Primary 20F14, 20F36, 52C35, 55P35, 14D99
DOI: https://doi.org/10.1090/S0002-9947-02-02948-3
Published electronically: January 11, 2002
MathSciNet review: 1881013
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as ``twistings" between the factors. The main homological results are given in terms of extensions of the ``infinitesimal braid relations" or ``universal Yang-Baxter Lie relations".


References [Enhancements On Off] (What's this?)

  • 1. M. Bendersky, and S. Gitler, The Cohomology of Certain Function Spaces, Trans. A.M.S., 326, No. 1, (1991), 423-440. MR 93d:55005
  • 2. V. Chari, and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, 1998, Cambridge, England. MR 95j:17010 (1st ed.)
  • 3. D. Cohen, F. R. Cohen, and M. Xicoténcatl, Lie Algebras Associated to Fiber-Type Arrangements, submitted.
  • 4. F. R. Cohen, The homology of iterated loop spaces, Springer-Verlag Lecture Notes in Mathematics, 533(1976), 207-351. MR 55i:9096
  • 5. F. R. Cohen, On genus one mapping class groups, function spaces, and modular forms, to appear in Cont. Math..
  • 6. F. R. Cohen, S. Gitler, Loop spaces of configuration spaces, braid-like groups, and knots, in Cohomological Methods in Homotopy Theory, Birkhäuser Verlag Progress in Mathematics vol. 196(2001), Basel, Berlin, Boston, 59-78.
  • 7. F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. 109(1979), 121-168. MR 80e:55024
  • 8. F. R. Cohen and T. Sato, On groups of homotopy groups, loop spaces, and braid-like groups, preprint.
  • 9. F. R. Cohen and L. R. Taylor, (a) On the representation theory associated to the cohomology of configuration spaces, Cont. Math. 146(1993), 91-109. MR 94i:57057 (b) Gelfand-Fuks cohomology, the cohomology of function spaces and the cohomology of configuration spaces, Springer-Verlag Lecture Notes in Mathematics, 657 (1979), 106-143. MR 80f:58050
  • 10. F. R. Cohen, and M. Xicoténcatl, On orbit configuration spaces associated to the Gaussian integers: homotopy groups, and homology groups, to appear in a Special Issue of Topology and its Applications devoted to the meeting ``Arrangements in Boston", to appear.
  • 11. V. G Drinfel'd, On quasi-triangular quasi-Hopf algebras, and a certain group closely connected with $Gal(\bar \mathbb Q/Q)$, Leningrad Math. J., 2(1991), 829-60. MR 92f:16047
  • 12. V. G. Drinfel'd, On the structure of quasi-triangular quasi-Hopf algebras, Funct. Anal. Appl. 26(1992), 63-5. MR 93i:16052
  • 13. E. Fadell and S. Husseini, (a) The space of loops on configuration spaces and the Majer-Terracini index, Topological Methods in Nonlinear Analysis, Journal of the Julius Schauder Center 11 (1998), 249-271. MR 2000a:55020 (b) Geometry and Topology of Configuration Spaces, Springer-Verlag, Springer Monographs in Mathematics, (2001). CMP 2001:06
  • 14. E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 119-126. MR 25:4537
  • 15. M. Falk, and R. Randell, The lower central series of a generalized pure braid arrangement, Invent. Math. 82 (1985),77-88. MR 88c:20048
  • 16. Y. Félix and J.-C. Thomas, Effet d'un attachement cellulaire dans l'homologie de l'espace des lacets, Ann. Inst. Fourier, Grenoble 39, 1(1989), 207-224. MR 99j:55012
  • 17. Y. Félix, and J.C. Thomas, Homologie des espaces de lacets des espaces de configuration, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 559-568. MR 95i:55007
  • 18. T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965) 295-322. MR 31:4033
  • 19. P. Hilton, On the homotopy groups of a union of spheres, Comment. Math. Helv. 29 (1955), 59-92. MR 16:1043d
  • 20. N. Jacobson, Lie algebras, Interscience Press, 1962. MR 26:1345
  • 21. T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Cont. Math. 78 (1988), 339-363. MR 90h:20056
  • 22. T. Kohno, Vassiliev invariants, and the de Rham complex on the space of knots, Cont. Math. 179(1994),123-138. MR 96g:57010
  • 23. I. Kriz, On the rational homotopy type of configuration spaces, Ann. of Math. 139(1994), 227-237. MR 95c:55012.
  • 24. J. Milnor, On the construction $F[K]$, London Mathematical Society Lecture Notes, 4(1972),119-136.
  • 25. J. Milnor, and J. Moore, On the structure of Hopf algebras, Ann. of Math., 81 (1965), 211-264. MR 30:4259
  • 26. J. Milnor and J. Stasheff, Characteristic classes, Ann. of Math. Studies 76, Princeton University Press, Princeton, 1974. MR 55:13428
  • 27. N. Steenrod, The Topology of Fibre Bundles, Princeton University Press, Princeton, 1951. MR 12:522b
  • 28. R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17-86. MR 15:890b
  • 29. B. Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996), 1057-1067. MR 97g:57033
  • 30. G. W. Whitehead, Elements of Homotopy Theory, Springer-Verlag Graduate Texts in Mathematics, 61(1978). MR 80b:55001
  • 31. M. Xicoténcatl, Orbit configuration spaces, infinitesimal braid relations, and equivariant function spaces, Trans. A.M.S., to appear.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20F14, 20F36, 52C35, 55P35, 14D99

Retrieve articles in all journals with MSC (2000): 20F14, 20F36, 52C35, 55P35, 14D99


Additional Information

F. R. Cohen
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
Email: cohf@math.rochester.edu

S. Gitler
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627, and Departamento de Matemáticas, Cinvestav, Apdo. Postal 14-740 , México, D.F. 07300
Email: sgitler@math.cinvestav.mx

DOI: https://doi.org/10.1090/S0002-9947-02-02948-3
Keywords: Braid groups, configuration spaces, descending central series, loop spaces
Received by editor(s): October 12, 1999
Received by editor(s) in revised form: September 1, 2001
Published electronically: January 11, 2002
Additional Notes: The authors were partially supported by the National Science Foundation Grant number 9704410.
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society