Moduli of suspension spectra

Klein, John

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

Moduli of suspension spectra
HTML articles powered by AMS MathViewer

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