On loop spaces of configuration spaces

Cohen, F.; Gitler, S.

doi:10.1090/S0002-9947-02-02948-3

On loop spaces of configuration spaces
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Trans. Amer. Math. Soc. 354 (2002), 1705-1748 Request permission

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

B. D. Gel′man, Generalized degree of multivalued mappings, Nonlinear operators in global analysis (Russian), Novoe Global. Anal., Voronezh. Gos. Univ., Voronezh, 1991, pp. 34–51, 161 (Russian). MR 1167015
Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632
D. Cohen, F. R. Cohen, and M. Xicoténcatl, Lie Algebras Associated to Fiber-Type Arrangements, submitted.
J. L. Walsh, On interpolation by functions analytic and bounded in a given region, Trans. Amer. Math. Soc. 46 (1939), 46–65. MR 55, DOI 10.1090/S0002-9947-1939-0000055-0
F. R. Cohen, On genus one mapping class groups, function spaces, and modular forms, to appear in Cont. Math..
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.
F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. (2) 109 (1979), no. 1, 121–168. MR 519355, DOI 10.2307/1971269
F. R. Cohen and T. Sato, On groups of homotopy groups, loop spaces, and braid-like groups, preprint.
Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
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.
V. G. Drinfel′d, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Algebra i Analiz 2 (1990), no. 4, 149–181 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 4, 829–860. MR 1080203
V. G. Drinfel′d, On the structure of quasitriangular quasi-Hopf algebras, Funktsional. Anal. i Prilozhen. 26 (1992), no. 1, 78–80 (Russian); English transl., Funct. Anal. Appl. 26 (1992), no. 1, 63–65. MR 1163026, DOI 10.1007/BF01077082
P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118. MR 141126, DOI 10.7146/math.scand.a-10517
Michael Falk and Richard Randell, The lower central series of generalized pure braid groups, Geometry and topology (Athens, Ga., 1985) Lecture Notes in Pure and Appl. Math., vol. 105, Dekker, New York, 1987, pp. 103–108. MR 873287
Hideo Takahashi, Self maps of suspension of sphere bundles over spheres, Math. J. Okayama Univ. 38 (1996), 155–158 (1998). MR 1644489
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 (French, with English and French summaries). MR 1296743, DOI 10.5802/aif.1409
T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295–322. MR 179791, DOI 10.1007/BF02566956
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
Toshitake Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 339–363. MR 975088, DOI 10.1090/conm/078/975088
Toshitake Kohno, Vassiliev invariants and de Rham complex on the space of knots, Symplectic geometry and quantization (Sanda and Yokohama, 1993) Contemp. Math., vol. 179, Amer. Math. Soc., Providence, RI, 1994, pp. 123–138. MR 1319605, DOI 10.1090/conm/179/01937
Igor Kříž, On the rational homotopy type of configuration spaces, Ann. of Math. (2) 139 (1994), no. 2, 227–237. MR 1274092, DOI 10.2307/2946581
J. Milnor, On the construction $F[K]$, London Mathematical Society Lecture Notes, 4(1972),119-136.
John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Burt Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996), no. 4, 1057–1067. MR 1404924, DOI 10.1016/0040-9383(95)00058-5
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508, DOI 10.1007/978-1-4612-6318-0
M. Xicoténcatl, Orbit configuration spaces, infinitesimal braid relations, and equivariant function spaces, Trans. A.M.S., to appear.