Moduli of suspension spectra

Klein, John

doi:10.1090/S0002-9947-04-03474-9

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a -connected spectrum , we study the moduli space of suspension spectra which come equipped with a weak equivalence to . 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.

[A-C-D] A. Adem, R. L. Cohen, and W. G. Dwyer, Generalized Tate homology, homotopy fixed points and the transfer, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 1–13. MR 1022669, https://doi.org/10.1090/conm/096/1022669
[A-K] Stephen T. Ahearn and Nicholas J. Kuhn, Product and other fine structure in polynomial resolutions of mapping spaces, Algebr. Geom. Topol. 2 (2002), 591–647. MR 1917068, https://doi.org/10.2140/agt.2002.2.591
[Ar] Greg Arone, A generalization of Snaith-type filtration, Trans. Amer. Math. Soc. 351 (1999), no. 3, 1123–1150. MR 1638238, https://doi.org/10.1090/S0002-9947-99-02405-8
[B-H] I. Berstein and P. J. Hilton, On suspensions and comultiplications, Topology 2 (1963), 73–82. MR 150775, https://doi.org/10.1016/0040-9383(63)90024-7
[Ba] Bauer, K. B.: On Hopf algebra type and rational calculus decompositions.
Ph.D. thesis, University of Illinois Urbana-Champaign 2001
[C] Gunnar Carlsson, Equivariant stable homotopy and Segal’s Burnside ring conjecture, Ann. of Math. (2) 120 (1984), no. 2, 189–224. MR 763905, https://doi.org/10.2307/2006940
[Go1] Thomas G. Goodwillie, Calculus. I. The first derivative of pseudoisotopy theory, 𝐾-Theory 4 (1990), no. 1, 1–27. MR 1076523, https://doi.org/10.1007/BF00534191
[Go2] Thomas G. Goodwillie, Calculus. II. Analytic functors, 𝐾-Theory 5 (1991/92), no. 4, 295–332. MR 1162445, https://doi.org/10.1007/BF00535644
[Go3] Goodwillie, T. G.: Calculus III: Taylor series. Geom. Topol. 7, 645-711 (2003).
[Gr] Brayton Gray, Desuspension at an odd prime, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 360–370. MR 764589, https://doi.org/10.1007/BFb0075577
[G-M] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178. MR 1230773, https://doi.org/10.1090/memo/0543
[Jo] Brenda Johnson, The derivatives of homotopy theory, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1295–1321. MR 1297532, https://doi.org/10.1090/S0002-9947-1995-1297532-6
[J-W] J. D. S. Jones and S. A. Wegmann, Limits of stable homotopy and cohomotopy groups, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 3, 473–482. MR 720798, https://doi.org/10.1017/S0305004100000864
[Kl1] John R. Klein, Poincaré duality embeddings and fiberwise homotopy theory, Topology 38 (1999), no. 3, 597–620. MR 1670412, https://doi.org/10.1016/S0040-9383(98)00034-2
[Kl2] John R. Klein, Axioms for generalized Farrell-Tate cohomology, J. Pure Appl. Algebra 172 (2002), no. 2-3, 225–238. MR 1906876, https://doi.org/10.1016/S0022-4049(01)00151-7
[Ku] Nicholas J. Kuhn, Suspension spectra and homology equivalences, Trans. Amer. Math. Soc. 283 (1984), no. 1, 303–313. MR 735424, https://doi.org/10.1090/S0002-9947-1984-0735424-1
[L-M-S] L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR 866482
[Mah] Mark Mahowald, The metastable homotopy of 𝑆ⁿ, Memoirs of the American Mathematical Society, No. 72, American Mathematical Society, Providence, R.I., 1967. MR 0236923
[Mc1] McCarthy, R.: On -excisive functors of module categories.
to appear in Math. Proc. Camb. Phil. Soc.
[Mc2] Randy McCarthy, Dual calculus for functors to spectra, Homotopy methods in algebraic topology (Boulder, CO, 1999) Contemp. Math., vol. 271, Amer. Math. Soc., Providence, RI, 2001, pp. 183–215. MR 1831354, https://doi.org/10.1090/conm/271/04357
[Mi] R. James Milgram, Unstable homotopy from the stable point of view, Lecture Notes in Mathematics, Vol. 368, Springer-Verlag, Berlin-New York, 1974. MR 0348740
[Qu] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
[Sc] Stefan Schwede, Spectra in model categories and applications to the algebraic cotangent complex, J. Pure Appl. Algebra 120 (1997), no. 1, 77–104. MR 1466099, https://doi.org/10.1016/S0022-4049(96)00058-8
[Wa] Friedhelm Waldhausen, Algebraic 𝐾-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR 802796, https://doi.org/10.1007/BFb0074449
[W-W] Michael Weiss and Bruce Williams, Automorphisms of manifolds and algebraic 𝐾-theory. II, J. Pure Appl. Algebra 62 (1989), no. 1, 47–107. MR 1026874, https://doi.org/10.1016/0022-4049(89)90020-0