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Moduli of suspension spectra


Author: John R. Klein
Translated by:
Journal: Trans. Amer. Math. Soc. 357 (2005), 489-507
MSC (2000): Primary 55P42, 55P43; Secondary 55P40, 55P65
DOI: https://doi.org/10.1090/S0002-9947-04-03474-9
Published electronically: March 23, 2004
MathSciNet review: 2095620
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Abstract | References | Similar Articles | Additional Information

Abstract: For a $1$-connected spectrum $E$, we study the moduli space of suspension spectra which come equipped with a weak equivalence to $E$. We construct a spectral sequence converging to the homotopy of the moduli space in positive degrees. In the metastable range, we get a complete homotopical classification of the path components of the moduli space. Our main tool is Goodwillie's calculus of homotopy functors.


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

  • [A-C-D] Adem, A., Cohen, R.L., Dwyer, W.G.: Tate homology, homotopy fixed points and the transfer.
    In: Algebraic Topology (Evanston 1988) Contemp. Math. 96, pp. 1-13.
    AMS 1989 MR 90k:55012
  • [A-K] Ahearn, S. T., Kuhn, N. J.: Product and other fine structure in polynomial resolutions of mapping spaces.
    Algebr. Geom. Topol. 2, 1123-1150 (2002) MR 2003j:55009
  • [Ar] Arone, G.: A generalization of the Snaith-type filtration.
    Trans. Amer. Soc. 351, 591-647 (1999) MR 99i:55011
  • [B-H] Berstein, I., Hilton, P. J.: On suspensions and comultiplications.
    Topology 2, 73-82 (1963) MR 27:762
  • [Ba] Bauer, K. B.: On Hopf algebra type and rational calculus decompositions.
    Ph.D. thesis, University of Illinois Urbana-Champaign 2001
  • [C] Carlsson, G.: Equvariant stable homotopy and Segal's Burnside ring conjecture.
    Ann. of Math. 120, 189-224 (1984) MR 86f:57036
  • [Go1] Goodwillie, T. G.: Calculus I, the derivative of a homotopy functor.
    $K$-theory 4, 1-27 (1990) MR 92m:57027
  • [Go2] Goodwillie, T. G.: Calculus II: analytic functors.
    $K$-theory 5, 295-332 (1992) MR 93i:55015
  • [Go3] Goodwillie, T. G.: Calculus III: Taylor series. Geom. Topol. 7, 645-711 (2003).
  • [Gr] Gray, B.: Desuspension at an odd prime.
    In: Algebraic topology, Aarhus 1982, Springer LNM 1051, pp. 360-370.
    Springer, 1984 MR 86d:55017
  • [G-M] Greenlees, J., May, J.: Generalized Tate Cohomology.
    (AMS MDual calculus for functors to spectra. emoirs, Vol. 543).
    AMS 1995 MR 96e:55006
  • [Jo] Johnson, B.: The derivatives of homotopy theory.
    Trans. Amer. Math. Soc. 347, 1295-1321 (1995) MR 96b:55012
  • [J-W] Jones, J.D.S., Wegmann S.A.: Limits of stable homotopy and cohomotopy groups.
    Math. Proc. Camb. Phil. Soc. 94, 473-482 (1983) MR 85b:55014
  • [Kl1] Klein, J. R.: Poincaré embeddings and fiberwise homotopy theory.
    Topology 38, 597-620 (1999) MR 2000b:57037
  • [Kl2] Klein, J. R.: Axioms for generalized Farrell-Tate cohomology.
    J. Pure Appl. Algebra 172(2-3), 225-238 (2002) MR 2003d:55006
  • [Ku] Kuhn, N. J.: Suspension spectra and homology equivalences.
    Trans. Amer. Math. Soc. 283, 303-313 (1984). MR 85g:55014
  • [L-M-S] Lewis, L. G., May, J. P., Steinberger, M., McClure, J. E.: Equivariant Stable Homotopy Theory.
    (LNM, Vol. 1213).
    Springer, 1986 MR 88e:55002
  • [Mah] Mahowald, M.: The metastable homotopy of $S^n$.
    (Memoirs of the Amer. Math. Soc, No. 72).
    Amer. Math. Soc. 1967 MR 38:5216
  • [Mc1] McCarthy, R.: On $n$-excisive functors of module categories.
    to appear in Math. Proc. Camb. Phil. Soc.
  • [Mc2] McCarthy, R.: Dual calculus for functors to spectra. Homotopy methods in algebraic topology (Boulder, CO 1999).
    (Contemp. Math. 271),
    Amer. Math. Soc. 2001 MR 2002c:18009
  • [Mi] Milgram, R. J.: Unstable Homotopy from the Stable Point of View.
    (LNM, Vol. 368),
    Springer, 1974 MR 50:1235
  • [Qu] Quillen, D.: Homotopical Algebra.
    (LNM, Vol. 43).
    Springer, 1967 MR 36:6480
  • [Sc] Schwede, S.: Spectra in model categories and applications to the algebraic cotangent complex.
    J. Pure Appl. Algebra 120, 77-104 (1997) MR 98h:55027
  • [Wa] Waldhausen, F.: Algebraic $K$-theory of spaces.
    In: Algebraic and geometric topology, Proceedings Rutgers 1983, LNM 1126, pp. 318-419.
    Springer, 1985 MR 86m:18011
  • [W-W] Weiss, M., Williams, B.: Automorphisms of manifolds and algebraic $K$-theory. II.
    J. Pure Appl. Algebra 62, 47-107 (1989) MR 91e:57055

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Additional Information

John R. Klein
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: klein@math.wayne.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03474-9
Received by editor(s): January 3, 2003
Received by editor(s) in revised form: July 1, 2003
Published electronically: March 23, 2004
Additional Notes: The author was partially supported by NSF Grant DMS-0201695
Article copyright: © Copyright 2004 American Mathematical Society

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