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Cohomological Methods in Homotopy Theory pp 185–216Cite as

Birkhäuser

New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces

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Part of the book series: Progress in Mathematics ((PM,volume 196))

Abstract

Let T(j) be the dual of the j th stable summand of Ω2 S 3 (at the prime 2) with top class in dimension j. Then it is known that T(j) is a retract of a suspension spectrum, and that the homotopy colimit of a certain sequence T(j)→ T(2j)→… is an infinite wedge of stable summands of K(V,1)’s, where V denotes an elementary abelian 2 group. In particular, when one starts with T(1), one gets K(Z/2, 1) = RP ∞ as one of the summands.

I discuss a generalization of this picture using higher iterated loopspaces and Eilenberg MacLane spaces. I consider certain finite spectra T(n, j) for n, j ≥ 0 (with T(1, j) = T(j)), dual to summands of Ωn+1 S N, conjecture generalizations of the above, and prove that these conjectures are correct in cohomology. So, for example, T(n,j) has unstable cohomology, and the cohomology of the hocolimit of a certain sequence T(n,j) → T(n, 2j) → …. agrees with the cohomology of the wedge of stable summands of K(V,n)’s corresponding to the wedge occurring in the n = 1 case above.

One can also map the T(n, j) to each other as n varies, and here the cohomological calculations imply a homotopical conclusion: the hocolimits that are nonzero, T(∞, 2k), for k ≥ 0, map to each other, giving rise to a fitration of H Z/2 which is equivalent to the mod 2 symmetric powers of spheres filtration.

Our homotopical constructions use Hopf invariant methods and loop-space technology. These are quite general and should be of independent interest. To study the action of the Steenrod operations on the cohomology of our spectra, we derive a Nishida formula for how X(Sq i) acts on Dyer—Lashof operations. This should be of use in other settings. In an appendix, we explain connections with recent work by Greg Arone and Mark Mahowald on the Goodwillie tower of the identity.

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References

  1. J. Adern, The relations on Steenrod powers of cohomology classes, Algebraic Geometry and Topology (R. H. Fox, et. al., editors), Princeton University Press, 1957, 191–242.

    Google Scholar 

  2. G. Z. Arone and W. G. Dwyer, Partition complexes, Tits buildings, and symmetric product spectra, preliminary preprint, 1997.

    Google Scholar 

  3. G. Arone and M. Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135(1999), 743–788.

    MathSciNet  MATH  Google Scholar 

  4. M. G. Barratt and H. R. Miller, On the antiautomorphism of the Steenrod algebra, A. M. S. Cont. Math. 12(1982), 47–52.

    Article  MATH  Google Scholar 

  5. E. H. Brown and R. L. Cohen, The Adams spectral sequence of Ω2S3 and BrownGitler spectra, Annals of Math. Study 113(1987), 101–125.

    MathSciNet  Google Scholar 

  6. R. R. Bruner, J. P. May, J. McClure, and M. Steinberger H∞ Ring Spectra and their Applications, Springer L. N. Math. 1176, 1986.

    Google Scholar 

  7. S. R. Bullett and I. G. MacDonald, On the Adern relations, Topology 21(1982), 329–332.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Carlsson, G. B. Segal’s Burnside ring conjecture for (Z/2)K,Topology 22(1983), 83–103.

    Article  MathSciNet  MATH  Google Scholar 

  9. F. R. Cohen, T. Lada, and J. P. May, The Homology of Iterated Loop Spaces, Springer L. N. Math. 533, 1976.

    Google Scholar 

  10. R. Fox and L. Neuwirth, The braid groups, Math. Scand. 10(1962), 119–126.

    MathSciNet  MATH  Google Scholar 

  11. P. Goerss, A direct construction for the duals of Brown-Gitler spectra, Ind. Math. J. 34(1985), 733–751.

    MathSciNet  MATH  Google Scholar 

  12. P. Goerss, J. Lannes, and F. Morel, Vecteurs de Witt non-commutatifs et representabilité de l’homologie modulo p, Invent. Math. 108(1992), 163–227.

    MathSciNet  MATH  Google Scholar 

  13. T. G. Goodwillie, Calculus III: the Taylor series of a hornotopy functor, preprint, c.1994.

    Google Scholar 

  14. H.-W. Henn, J. Lannes, and L. Schwartz The categories of unstable modules and unstable algebras modulo nilpotent objects, Amer. J. Math. 115(1993), 1053–1106.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. J. Hunter and N. J. Kuhn, Characterizations of spectra with Lf —injective cohomology which satisfy the Brown—Gitler property, to appear in Trans. A.M.S.

    Google Scholar 

  16. D. Husemoller, Fibre Bundles, Springer Graduate Texts in Math. 20, 1966.

    Google Scholar 

  17. P. Krason and N. J. Kuhn, On embedding polynomial functors in symmetric powers, J. Algebra. 99, (1994)281–294.

    Article  MathSciNet  Google Scholar 

  18. N. J. Kuhn, The geometry of the James Hopf maps, Pac. J. Math. 102(1982), 397–412.

    Article  MATH  Google Scholar 

  19. N. J. Kuhn, A Kahn-Priddy sequence and a conjecture of G. W. Whitehead, Math. Proc. Camb Phil. Soc. 92(1982), 467–483.

    Article  MATH  Google Scholar 

  20. N. J. Kuhn, Spacelike resolutions of spectra, A.M.S. Cont. Math. Series 19(1983), 153–165.

    MATH  Google Scholar 

  21. N. J. Kuhn, Suspension spectra and homology equivalences, Trans. A. M. S. 283(1984), 303–313.

    Article  MATH  Google Scholar 

  22. N. J. Kuhn, The transfer and the James-Hopf invariants, Math. Zeit. 196(1987), 391–405.

    Article  MATH  Google Scholar 

  23. N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: I, Amer. J. Math. 116(1994), 327–360.

    Article  MathSciNet  MATH  Google Scholar 

  24. N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: II, K-Theory J. 8(1994), 395–428.

    Article  MATH  Google Scholar 

  25. N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: III, K-Theory J. 9(1995), 273–303.

    Article  MATH  Google Scholar 

  26. N. J. Kuhn, Computations in generic representation theory: maps from symmetric powers to composite functors, Trans. A.M.S. 350(1998), 4221–4233.

    Article  MATH  Google Scholar 

  27. J. Lannes, Sur le n dual du n-ème spectre de Brown-Gitler, Math. Zeit. 159(1988), 29–42.

    Article  MathSciNet  Google Scholar 

  28. J. Lannes,Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire, Pub. I.H.E.S. 75(1992), 135–244.

    MathSciNet  MATH  Google Scholar 

  29. J. Lannes and L.Schwartz, Sur la structure des A-modules instables injectifs,Topology 28(1989), 153–169.

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Lannes and S. Zarati, Sur les Lf-injectifs, Ann. Scient. Ec. Norm. Sup 19(1986), 1–31.

    MathSciNet  Google Scholar 

  31. J. Lannes and S. Zarati, Sur les foncteurs dérivés de la déstablisation, Math. Zeit. 194(1987), 25–59.

    Article  MathSciNet  MATH  Google Scholar 

  32. L. G. Lewis, J. P. May, J. McClure, and M. Steinberger Equivariant Stable Homotopy Theory, Springer L. N. Math. 1213, 1986.

    Google Scholar 

  33. M. Mahowald, A new infinite family in 271-’, Topology 16(1977), 249–256.

    Article  MathSciNet  MATH  Google Scholar 

  34. J. P. May, The Geometry of Iterated Loop Spaces, Springer L. N. Math. 271, 1972.

    Google Scholar 

  35. C. A. McGibbon, Infinite loop spaces and Neisendorfer localization, Proc. A.M.S. 125(1997), 309–313.

    Article  MathSciNet  MATH  Google Scholar 

  36. H. R. Miller, An algebraic analogue of a conjecture of G. W. Whitehead, Proc. A.M.S. 84(1982), 131–137.

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  38. S. A. Mitchell and S. B. Priddy, Stable splittings derived from the Steinberg module, Topology 22(1983), 285–298.

    Article  MathSciNet  MATH  Google Scholar 

  39. N. Popescu, Abelian Categories with Applications to Rings and Modules, Academic Press, London, 1973.

    MATH  Google Scholar 

  40. L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point conjecture, Chicago Lectures in Math., Unversity of Chicago Press, 1994.

    Google Scholar 

  41. W. M. Singer, A new chain complex for the homology of the Steenrod algebra, Math. Proc. Camb. Phil. Soc. 90(1981), 279–292.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA

    Nicholas J. Kuhn

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  1. Nicholas J. Kuhn
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Editors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    Jaume Aguadé , Carles Broto  & Carles Casacuberta ,  & 

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Kuhn, N.J. (2001). New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces. In: Aguadé, J., Broto, C., Casacuberta, C. (eds) Cohomological Methods in Homotopy Theory. Progress in Mathematics, vol 196. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8312-2_14

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  • DOI: https://doi.org/10.1007/978-3-0348-8312-2_14

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9513-2

  • Online ISBN: 978-3-0348-8312-2

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