## Moduli of suspension spectra

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- by John R. Klein PDF
- Trans. Amer. Math. Soc.
**357**(2005), 489-507 Request permission

## 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

- 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**, DOI 10.1090/conm/096/1022669 - 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**, DOI 10.2140/agt.2002.2.591 - Greg Arone,
*A generalization of Snaith-type filtration*, Trans. Amer. Math. Soc.**351**(1999), no. 3, 1123–1150. MR**1638238**, DOI 10.1090/S0002-9947-99-02405-8 - I. Berstein and P. J. Hilton,
*On suspensions and comultiplications*, Topology**2**(1963), 73–82. MR**150775**, DOI 10.1016/0040-9383(63)90024-7 - Gunnar Carlsson,
*Equivariant stable homotopy and Segal’s Burnside ring conjecture*, Ann. of Math. (2)**120**(1984), no. 2, 189–224. MR**763905**, DOI 10.2307/2006940 - Thomas G. Goodwillie,
*Calculus. I. The first derivative of pseudoisotopy theory*, $K$-Theory**4**(1990), no. 1, 1–27. MR**1076523**, DOI 10.1007/BF00534191 - Thomas G. Goodwillie,
*Calculus. II. Analytic functors*, $K$-Theory**5**(1991/92), no. 4, 295–332. MR**1162445**, DOI 10.1007/BF00535644 - 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**, DOI 10.1007/BFb0075577 - J. P. C. Greenlees and J. P. May,
*Generalized Tate cohomology*, Mem. Amer. Math. Soc.**113**(1995), no. 543, viii+178. MR**1230773**, DOI 10.1090/memo/0543 - Brenda Johnson,
*The derivatives of homotopy theory*, Trans. Amer. Math. Soc.**347**(1995), no. 4, 1295–1321. MR**1297532**, DOI 10.1090/S0002-9947-1995-1297532-6 - 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**, DOI 10.1017/S0305004100000864 - John R. Klein,
*Poincaré duality embeddings and fiberwise homotopy theory*, Topology**38**(1999), no. 3, 597–620. MR**1670412**, DOI 10.1016/S0040-9383(98)00034-2 - John R. Klein,
*Axioms for generalized Farrell-Tate cohomology*, J. Pure Appl. Algebra**172**(2002), no. 2-3, 225–238. MR**1906876**, DOI 10.1016/S0022-4049(01)00151-7 - Nicholas J. Kuhn,
*Suspension spectra and homology equivalences*, Trans. Amer. Math. Soc.**283**(1984), no. 1, 303–313. MR**735424**, DOI 10.1090/S0002-9947-1984-0735424-1 - 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**, DOI 10.1007/BFb0075778 - Mark Mahowald,
*The metastable homotopy of $S^{n}$*, Memoirs of the American Mathematical Society, No. 72, American Mathematical Society, Providence, R.I., 1967. MR**0236923** - 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**, DOI 10.1090/conm/271/04357 - 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**, DOI 10.1007/BFb0070455 - Daniel G. Quillen,
*Homotopical algebra*, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR**0223432**, DOI 10.1007/BFb0097438 - 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**, DOI 10.1016/S0022-4049(96)00058-8 - 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**, DOI 10.1007/BFb0074449 - 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**, DOI 10.1016/0022-4049(89)90020-0

## Additional Information

**John R. Klein**- Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
- MR Author ID: 308817
- Email: klein@math.wayne.edu
- 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
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2095620