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Homotopy Lie groups


Author: Jesper M. Møller
Journal: Bull. Amer. Math. Soc. 32 (1995), 413-428
MSC: Primary 55R35; Secondary 55P35
DOI: https://doi.org/10.1090/S0273-0979-1995-00613-0
MathSciNet review: 1322786
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Abstract: Homotopy Lie groups, recently invented by W.G. Dwyer and C.W. Wilkerson [13], represent the culmination of a long evolution. The basic philosophy behind the process was formulated almost 25 years ago by Rector [32, 33] in his vision of a homotopy theoretic incarnation of Lie group theory. What was then technically impossible has now become feasible thanks to modern advances such as Miller's proof of the Sullivan conjecture [25] and Lannes's division functors [22]. Today, with Dwyer and Wilkerson's implementation of Rector's vision, the tantalizing classification theorem seems to be within grasp.

Supported by motivating examples and clarifying exercises, this guide quickly leads, without ignoring the context or the proof strategy, from classical finite loop spaces to the important definitions and striking results of this new theory.


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  • [1] J.C. Becker and D.H. Gottlieb, Transfer maps for fibrations and duality, Comp. Math. 33 (1976), 107-133. MR 0436137 (55:9087)
  • [2] D.J. Benson, Polynomial invariants of finite groups, London Math. Soc. Lecture Note Ser., vol. 190, Cambridge Univ. Press, Cambridge, 1994. MR 1249931 (94j:13003)
  • [3] A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115-207. MR 0051508 (14:490e)
  • [4] A.K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133-150. MR 0380779 (52:1676)
  • [5] A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localizations, 2nd ed., Lecture Notes in Math., vol. 304, Springer-Verlag, Berlin, Heidelberg, and New York, 1987. MR 0365573 (51:1825)
  • [6] A. Clark and J.R. Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974), 425-434. MR 0367979 (51:4221)
  • [7] W.G. Dwyer, H.R. Miller, and C.W. Wilkerson, The homotopy uniqueness of $ {BS^{3}}$, Algebraic Topology (Barcelona, 1986) (J. Aguadé and R. Kane, eds.), Lecture Notes in Math., vol. 1298, Springer-Verlag, Berlin, Heidelberg, and New York, 1987, pp. 90-105. MR 928825 (89e:55019)
  • [8] -, Homotopical uniqueness of classifying spaces, Topology 31 (1992), 29-45. MR 1153237 (92m:55013)
  • [9] W.G. Dwyer and C.W. Wilkerson, Smith theory and the functor T, Comment. Math. Helv. 66 (1991), 1-17. MR 1090162 (92i:55006)
  • [10] -, A cohomology decomposition theorem, Topology 31 (1992), 433-443. MR 1167181 (93h:55008)
  • [11] -, The center of a p-compact group, The Čech Centennial: A Conference on Homotopy Theory (M. Cenkl and H. Miller, eds.), Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 119-157. MR 1320990 (96a:55024)
  • [12] -, A new finite loop space at the prime 2, J. Amer. Math. Soc. 6 (1993), 37-64. MR 1161306 (93d:55011)
  • [13] -, Homotopy fixed point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), 395-442. MR 1274096 (95e:55019)
  • [14] -, Product splittings for p-compact groups, Preprint, 1994.
  • [15] W.G. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic Topology (Barcelona, 1986) (J. Aguadé and R. Kane, eds.), Lecture Notes in Math., vol. 1298, Springer-Verlag, Berlin, Heidelberg, and New York, 1987, pp. 106-119. MR 928826 (89b:55018)
  • [16] S. Jackowski, J. McClure, and R. Oliver, Homotopy classification of self-maps of BG via G-actions, Part I, Ann. of Math. (2) 135 (1992), 183-226. MR 1147962 (93e:55019a)
  • [17] -, Homotopy classification of self-maps of BG via G-actions, Part II, Ann. of Math. (2) 135 (1992), 227-270. MR 1154593 (93e:55019b)
  • [18] S. Jackowski and J.E. McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups, Topology 31 (1992), 113-132. MR 1153240 (92k:55026)
  • [19] A. Jeanneret and U. Suter, Réalisation topologique de certaines algèbres associées aux algèbres de Dickson, Algebraic Topology, Homotopy and Group Cohomology (Proceedings, Barcelona, 1990) (F. R. Cohen, J. Aguadé, and M. Castellet, eds.), Lecture Notes in Math., vol. 1509, Springer-Verlag, Berlin, Heidelberg, and New York, 1992, pp. 235-239. MR 1185973 (93k:55022)
  • [20] D.M. Kan and W.P. Thurston, Every connected space has the homology of a $ {K(\pi, 1)}$, Topology 15 (1976), 253-258. MR 0413089 (54:1210)
  • [21] R. Kane, The homology of Hopf spaces, North-Holland Math. Library, vol. 40, Elsevier, Amsterdam, 1988. MR 961257 (90f:55018)
  • [22] J. Lannes, Sur les espaces fonctionnels dont la source est la classifiant d'un p-group abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135-244. MR 1179079 (93j:55019)
  • [23] -, Theorie homotopique des groupes de Lie [d'aprés W.G. Dwyer and C.W. Wilkerson], Sém. Bourbaki 776 (1993), 1-23.
  • [24] J. Lannes and S. Zarati, Théorie de Smith algébrique et classification des $ {H^{\ast}V - \mathcal{U}}$ -injectifs, Preprint.
  • [25] H.R. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), 39-87. MR 750716 (85i:55012)
  • [26] J.W. Milnor and J.C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. MR 0174052 (30:4259)
  • [27] J.M. Møller, Completely reducible p-compact groups. The Čech Centennial: A Conference on Homotopy Theory (M. Cenkl and H. Miller, eds.), Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 369-383. MR 1321001 (97b:55020)
  • [28] -, Rational isomorphisms of p-compact groups, Topology (to appear). MR 1367281 (97b:55019)
  • [29] J.M. Møller and D. Notbohm, Centers and finite coverings of finite loop spaces, J. Reine Angew. Math. 456 (1994), 99-133. MR 1301453 (95j:55029)
  • [30] D. Notbohm, Unstable splittings of classifying spaces of p-compact groups, Preprint, 1994. MR 1765793 (2001d:55004)
  • [31] D. Rector, Loop structures on the homotopy type of $ {S^3}$, Symposium on Algebraic Topology (P. Hilton, ed.), Lecture Notes in Math., vol. 249, Springer-Verlag, Berlin, Heidelberg, and New York, 1971, pp. 99-105. MR 0339153 (49:3916)
  • [32] -, Subgroups of finite dimensional topological groups, J. Pure Appl. Algebra 1 (1971), 253-273. MR 0301734 (46:889)
  • [33] D. Rector and J. Stasheff, Lie groups from a homotopy point of view, Localization in Group Theory and Homotopy Theory and Related Topics (P. Hilton, ed.), Lecture Notes in Math., vol. 418, Springer-Verlag, Berlin, Heidelberg, and New York, 1974, pp. 121-131. MR 0377868 (51:14037)
  • [34] L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point conjecture, Univ. of Chicago Press, Chicago, IL, 1994. MR 1282727 (95d:55017)
  • [35] L. Smith and R.M. Switzer, Realizability and nonrealizability of Dickson algebras, Proc. Amer. Math. Soc. 89 (1983), 303-313. MR 712642 (85e:55036)
  • [36] H. Toda, Cohomology of the classifying spaces of exceptional Lie groups, Manifolds Tokyo 1973 (A. Hattori, ed.), Univ. of Tokyo Press, Tokyo, 1975, pp. 265-271. MR 0368059 (51:4301)

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Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1995-00613-0
Keywords: Finite loop space, p-compact group, maximal torus, Weyl group
Article copyright: © Copyright 1995 American Mathematical Society

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