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Homotopy Theory of Classifying Spaces of Compact Lie Groups

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Algebraic Topology and Its Applications

Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 27))

Abstract

The basic problem of homotopy theory is to classify spaces and maps between spaces, up to homotopy, by means of invariants like cohomology. In the last decade some striking progress has been made with this problem when the spaces involved are classifying spaces of compact Lie groups. For example, it has been shown, for G connected and simple, that if two self maps of BG agree in rational cohomology then they are homotopic. It has also been shown that if a space X has the same mod p cohomology, cup product, and Steenrod operations as a classifying space BG then (at least if p is odd and G is a classical group) X is actually homotopy equivalent to BG after mod p completion. Similar methods have also been used to obtain new results on Steenrod’s problem of constructing spaces with a given polynomial cohomology ring. The aim of this paper is to describe these results and the methods used to prove them.

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References

  1. J.F. Adams, Maps between classifying spaces, II, Inventiones math. 49 (1978), 1–65.

    Article  MATH  Google Scholar 

  2. J.F. Adams & Z. Mahmud, Maps between classifying spaces, Inventiones math. 35 (1976), 1–41.

    Article  MATH  MathSciNet  Google Scholar 

  3. J F Adams & Z. Mahmud, Maps between classifying spaces, III,Topological topics, Cambridge Univ. Press (1983), 136–153.

    Google Scholar 

  4. J.F. Adams & C. Wilkerson, Finite H-spaces and algebras over the Steenrod algebra,Annals of Math. 111 (1980), 95–143.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Aguade, Constructing modular classifying spaces, Israel Jour. Math. 66 (1989), 23–40.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts,Ann. Math. 57 (1953), 115–207.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Bousfield, Homotopy spectral sequences and obstructions, Israel J. Math. 66 (1989), 54–104.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Bousfield & D. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer-Verlag (1972).

    Google Scholar 

  9. G. Bredon, Sheaf theory, McGraw-Hill, 1967.

    Google Scholar 

  10. G. Carlsson, Equivariant stable homotopy and Sullivan’s conjecture, Inventiones math. (to appear).

    Google Scholar 

  11. H. Cartan & S. Eilenberg, Homological algebra, Princeton Univ. Press, 1965.

    Google Scholar 

  12. A. Clark & J. Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974), 425–434.

    MATH  MathSciNet  Google Scholar 

  13. A. Dress, Induction and structure theorems for orthogonal representations of finite groups, Annals of Math. 102 (1975), 291–325.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Dwyer & D. Kan, Realizing diagrams in the homotopy category by means of diagrams of simplicial sets, Proc. Amer. Math. Soc. 91 (1984), 456–460.

    Article  MATH  MathSciNet  Google Scholar 

  15. W. Dwyer & D. Kan, Centric maps and realization of diagrams in the homotopy category, (preprint).

    Google Scholar 

  16. W. Dwyer, H. Miller, & J. Neisendorfer, Fiberwise completion and unstable Adams spectral sequences,Israel J. Math. 66 (1989), 160–178.

    Article  MATH  MathSciNet  Google Scholar 

  17. W. Dwyer, H. Miller, & C. Wilkerson, The homotopic uniqueness of BS 3, Algebraic topology, Barcelona, 1976, Lecture Notes in Math. 1298, Springer-Verlag (1987), 90–105.

    Google Scholar 

  18. W. Dwyer, H. Miller, & C. Wilkerson, Homotopical uniqueness of classifying spaces, (preprint).

    Google Scholar 

  19. W. Dwyer & G. Mislin, On the homotopy type of the components of map, (BS 3 BS 3 ), Algebraic topology, Barcelona, 1986, Lecture Notes in Math. 1298, Springer-Verlag (1987), 82–89.

    Google Scholar 

  20. W. Dwyer & C. Wilkerson, A cohomology decomposition theorem, (preprint).

    Google Scholar 

  21. W. Dwyer & C. Wilkerson, A new finite loop space at the prime two, (preprint).

    Google Scholar 

  22. W. Dwyer & A. Zabrodsky, Maps between classifying spaces, Algebraic topology, Barcelona, 1986, Lecture Notes in Math. 1298; Springer-Verlag (1987), 106–119.

    Google Scholar 

  23. M. Feshbach, The Segal conjecture for compact Lie groups, Topology 26 (1987), 1–20.

    Article  MATH  MathSciNet  Google Scholar 

  24. E. Friedlander, Unstable K-theories of the algebraic closure of a finite field, Comment. Math. Helv. 50 (1975), 145–154.

    Article  MATH  MathSciNet  Google Scholar 

  25. E. Friedlander, Exceptional isogenies and the classifying spaces of simple Lie groups, Annals of Math. 101 (1975), 510–520.

    Article  MATH  Google Scholar 

  26. E. Friedlander & G. Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups, Comment. Math. Helv. 59 (1984), 347–361.

    Article  MATH  MathSciNet  Google Scholar 

  27. E. Friedlander & G. Mislin, Locally finite approximations of Lie groups. I, Inventiones math. 83 (1986), 425–436.

    Article  MATH  MathSciNet  Google Scholar 

  28. E. Friedlander & G. Mislin, Locally finite approximations of Lie groups. II, Math. Proc. Camb. Phil. Soc. 100 (1986), 505–517.

    Article  MATH  MathSciNet  Google Scholar 

  29. P. Gabriel & M. Zisman, Calculus of fractions and homotopy theory, Springer-Verlag, 1967.

    Google Scholar 

  30. P. Hilton & U. Stammbach, A course in homological algebra, Springer-Verlag,1971.

    Google Scholar 

  31. M. Hopkins, Thesis, Oxford Univ. (1984).

    Google Scholar 

  32. J. Hubbuck, Homotopy representations of Lie groups, New developments in topology, London Math. Soc. Lecture Notes 11, Cambridge Univ. Press (1974), 33–41.

    Google Scholar 

  33. J. Hubbuck, Mapping degrees for classifying spaces I, Quarterly Journal of Math. Oxford Ser. (2) 25 (1974), 113–133.

    Article  MATH  MathSciNet  Google Scholar 

  34. W. Hurewicz, Beiträge zur Topologie der Deformationen IV: Asphärische Räume, Proc. Akad. Wetensch. Amsterdam 39 (1936), 215–224.

    MATH  Google Scholar 

  35. K. Ishiguro, Unstable Adams operations on classifying spaces, Math. Proc. Camb. Phil. Soc. 102 (1987), 71–75.

    Article  MATH  MathSciNet  Google Scholar 

  36. S. Jackowski & J. McClure, Homotopy approximations for classifying spaces of compact Lie groups, Proceedings of Arcata conference in algebraic topology, Lecture Notes in Math. 1370; Springer-Verlag (1989), 221–234.

    Google Scholar 

  37. S. Jackowski & J. McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups, Topology (to appear).

    Google Scholar 

  38. S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-maps of BG via G-actions, Annals of Math. (to appear).

    Google Scholar 

  39. S. Jackowski, J. McClure, & B. Oliver, Maps between classifying spaces revisited, (in preparation).

    Google Scholar 

  40. S. Jackowski, J. McClure, & B. Oliver, Self homotopy equivalences of BG, (in preparation).

    Google Scholar 

  41. J. Lannes, Sur la cohomologie modulo p des p-groupes abéliens élémentaires, Homotopy theory, Proceedings of the Durham symposium 1985, Cambridge Univ. Press (1987), 97–116.

    Google Scholar 

  42. J. Lannes, Sur les espaces fonctionnels dont la source est la classifiant d’un p-groupe abélien élémentaire, (preprint).

    Google Scholar 

  43. H. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. Math. 120 (1984), 39–87.

    Article  MATH  Google Scholar 

  44. J. Milnor, On axiomatic homology theory, Pac. J. Math. 12 (1962), 337–341.

    MATH  MathSciNet  Google Scholar 

  45. G. Mislin, The homotopy classification of self-maps of infinite quaternionic projective space, Quarterly J. Math. Oxford 38 (1987), 245–257.

    Article  MATH  MathSciNet  Google Scholar 

  46. G. Mislin & C. Thomas, On the homotopy set [Br, BG] with it finite and G a compact connected Lie group„ Quarterly J. Math. Oxford 40 (1989), 65–78.

    Article  MATH  MathSciNet  Google Scholar 

  47. D. Notbohm, Abbildungen zwischen klassifizierenden Räume, Dissertation, Göttingen (1988).

    Google Scholar 

  48. R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), 79–96.

    Article  MATH  MathSciNet  Google Scholar 

  49. R. Oliver, A proof of the Conner conjecture, Ann. Math. 103 (1976), 637–644.

    Article  MATH  MathSciNet  Google Scholar 

  50. L. Puig, Structure locale dans les groups finis, Bull. Soc. Math. France, Mémoire 47 (1976).

    Google Scholar 

  51. D. Quillen, The spectrum of an equivariant cohomology ring I, Ann. Math. 94 (1971), 549–572.

    Article  MATH  MathSciNet  Google Scholar 

  52. D. Quillen, 55,E-586, Ann. Math. 96 (1972).

    Google Scholar 

  53. D. Rector, Loop structure on the homotopy type of S31, Symposium on algebraic topology, Lecture Notes in Math. 249, Springer-Verlag (1971), 99–105.

    Google Scholar 

  54. G. Segal, Classifying spaces and spectral sequences, Publ. Math. I.H.E.S. 34 (1968), 105–112.

    Google Scholar 

  55. J. Slominska, Homotopy colimits on E-I-categories, Algebraic topology, Poznan, 1989, Lecture Notes in Math., Springer-Verlag (to appear).

    Google Scholar 

  56. D. Sullivan, Geometric topology, Part I: Localization, periodicity and Galois symmetry,Mimeographed notes, M.I.T. (1970).

    Google Scholar 

  57. C.W. Wilkerson, Self-maps of classifying spaces, Localization in group theory and homotopy theory,Lecture Notes in Math. 418; Springer-Verlag (1974), 150–157.

    Google Scholar 

  58. Z. Wojtkowiak, On maps from holim F to Z, Algebraic topology, Barcelona, 1986, Lecture Notes in Math. 1298; Springer-Verlag (1987), 227–236.

    Google Scholar 

  59. A. Zabrodsky, On the realization of invariant subgroups of π * X, Trans. Amer. Math. Soc. 285 (1984), 467–496.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. S.C.F, Inst. of Math., University of Warsaw, Banacha 2, PL02-097, Warsaw, Poland

    Stefan Jackowski, James McClure, Bob Oliver & Bob Oliver

  2. R. O., UER de Mathematics, Université de Paris VII, Tour 45-55, 2 place Jussieu, 75251, Paris, Cedex 05, France

    Stefan Jackowski & James McClure

Authors
  1. Stefan Jackowski
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  2. James McClure
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  3. Bob Oliver
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Stanford University, 94305, Stanford, CA, USA

    Gunnar E. Carlsson  & Ralph L. Cohen  & 

  2. Department of Mathematics, Princeton University, 08544, Princeton, NJ, USA

    Wu-Chung Hsiang

  3. Mathematics Research Center, University of Warwick, CV4 7AL, Coventry, UK

    John D. S. Jones

  4. Mathematical Sciences Research Institute, 1000 Centennial Drive, 94720, Berkeley, CA, USA

    John D. S. Jones

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© 1994 Springer-Verlag New York, Inc.

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Jackowski, S., McClure, J., Oliver, B. (1994). Homotopy Theory of Classifying Spaces of Compact Lie Groups. In: Carlsson, G.E., Cohen, R.L., Hsiang, WC., Jones, J.D.S. (eds) Algebraic Topology and Its Applications. Mathematical Sciences Research Institute Publications, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9526-3_4

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  • DOI: https://doi.org/10.1007/978-1-4613-9526-3_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9528-7

  • Online ISBN: 978-1-4613-9526-3

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