×

zbMATH — the first resource for mathematics

The fibre of the iterated Freudenthal suspension. (English) Zbl 0793.55006
A classical construction in Algebraic Topology is the iterated Freudenthal suspension map which induces the suspension homomorphism on homotopy groups. In this paper, we give combinatorially defined constructions for the homotopy theoretic fibre of the iterated loops of the iterated Freudenthal suspension map. We further prove that these fibres admit stable decompositions. As an application, we obtain in the case of spheres certain cofibre sequences when localized at a single prime. In the case of the single suspension and at the prime 2, it reduces to a classical cofibre sequence involving Brown-Gitler spectra.
Reviewer: S.Wong (Rochester)
MSC:
55P40 Suspensions
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [AK] Araki, S., Kudo, T.: Topology ofH n -spaces andH-squaring operations. Mem. Fac. Sci. Kyusyu Univ., Ser. A 85-120 (1956)
[2] [A] Arkowitz, M.: The generalized Whitehead product. Pac. J. Math.12, 7–23 (1962) · Zbl 0118.18404
[3] [BV] Boardman, J.M., Vogt, R.M.: Homotopy-everythingH-spaces. Bull. Am. Math. Soc.74, 1117–1122 (1968) · Zbl 0165.26204
[4] [B] Browder, W.: Homology operations and loop spaces. Ill. J. Math.4, 347–357 (1960) · Zbl 0107.40404
[5] [BC] Brown, E.H., Jr., Cohen, R.L.: The Adams spectral sequence of{\(\Omega\)} 2 S 3 and Brown Gitler spectra. Ann. Math. Stud.113, 101–125 (1987)
[6] [BG] Brown, E.H., Jr., Gitler, S.: A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra. Topology12, 283–296 (1973) · Zbl 0266.55012
[7] [C1] Cohen, F.R.: The unstable decomposition of{\(\Omega\)} 2 2 X and its application. Math. Z.182, 553–568 (1983) · Zbl 0505.55007
[8] [C2] Cohen, F.R.: A course in some aspects of classical homotopy theory. In: Miller, M.R., Ravenel, D.C. (eds.) Algebraic Topology. (Lect. Notes Math., vol. 1286, pp. 1–92) Berlin Heidelberg New York: Springer 1985
[9] [CLM] Cohen, F.R., Lada, T.J., May, J.P.: The Homology of Iterated Loop Spaces. (Lect. Notes Math., vol. 533) Berlin Heidelberg New York: Springer 1976 · Zbl 0334.55009
[10] [CMM] Cohen, F.R., Mahowald, M.E., Milgram, R.J.: The stable decomposition of the double loop space of sphere. In: Milgram, R.J. (ed.) Algebraic and Geometric Topology. (Proc. Symp. Pure Math., vol. 32, pp. 225–228) Providence, RI: Am. Math. Soc. 1978
[11] [CMT] Cohen, F.R., May, J.P., Taylor, L.R.: Splitting of certain spacesCX, Math. Proc. Camb. Philos. Soc.84, 465–496 (1978) · Zbl 0408.55006
[12] [CT] Cohen, F.R., Taylor, L.R.: The topology of configuration spaces. (Preprint 1990)
[13] [DL] Dyer, E., Lashof, R.: Homology of iterated loop spaces. Am. J. Math.84, 35–88 (1962) · Zbl 0119.18206
[14] [DT] Dold, A., Thom, R.: Quasifaserungen und unendliche symmetrische Produkte. Ann. Math.67, 239–281 (1958) · Zbl 0091.37102
[15] [FN] Fadell, E., Neuwirth, L.: Configuration spaces. Math. Scand.10, 119–126 (1962) · Zbl 0136.44104
[16] [G] Gray, B.: On the iterated suspension, Topology27, 301–310 (1988) · Zbl 0668.55005
[17] [HMR] Hilton, J.P., Mislin, G., Roitberg, J.: Localization of Nilpotent Groups and Spaces. Amsterdam: North-Holland 1975 · Zbl 0323.55016
[18] [Ma] Mahowald, M.: A new infinite family in2{\(\pi\)} * S . Topology16, 249–256 (1977) · Zbl 0357.55020
[19] [M] May, J.P.: The Geometry of Iterated Loop Spaces. (Lect. Notes Math., vol. 271) Berlin Heidelberg New York: Springer 1972 · Zbl 0244.55009
[20] [MT] May, J.P., Taylor, L.R.: Generalizd splitting theorems. Math. Proc. Camb. Philos. Soc.93, 73–86 (1983) · Zbl 0519.55005
[21] [Mi] Milgram, R.J.: Iterated loop spaces. Ann. Math.84, 386–403 (1966) · Zbl 0145.19901
[22] [Sn] Snaith, V.P.: A stable decomposition of{\(\Omega\)} n n X. J. Lond. Math. Soc.7, 577–583 (1974) · Zbl 0275.55019
[23] [S] Steenrod, N.: A conveneint category of topological spaces. Mich. Math. J.14, 133–152 (1967) · Zbl 0145.43002
[24] [W] Whitehead, G.W.: Elements of Homotopy Theory. Berlin Heidelberg New York: Springer 1978 · Zbl 0406.55001
[25] [Wh] Whitney, H.: Complex Analytic Varieties. Reading London: Addison-Wesley 1972 · Zbl 0265.32008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.