$EHP$ spectra and periodicity. I. Geometric constructions
HTML articles powered by AMS MathViewer
- by Brayton Gray
- Trans. Amer. Math. Soc. 340 (1993), 595-616
- DOI: https://doi.org/10.1090/S0002-9947-1993-1152323-X
- PDF | Request permission
Abstract:
The techniques used in $EHP$ calculation are studied, and lead to the notion of an $EHP$ spectrum. A simple inductive procedure suggests the existence of higher order $EHP$ spectra in which the first differential corresponds to ${v_n}$ multiplication. The next case $(n = 1)$ is constructed using the work of Cohen, Moore, and Neisendorfer. Some of the expected universal properties are proven.References
- David Anick, Differential algebras in topology, Research Notes in Mathematics, vol. 3, A K Peters, Ltd., Wellesley, MA, 1993. MR 1213682 D. Anick and B. Gray, Small $H$ spaces related to Moore spaces, Topology (submitted).
- M. G. Barratt, Note on a formula due to Toda, J. London Math. Soc. 36 (1961), 95โ96. MR 125582, DOI 10.1112/jlms/s1-36.1.95 โ, private communication.
- A. K. Bousfield and E. B. Curtis, A spectral sequence for the homotopy of nice spaces, Trans. Amer. Math. Soc. 151 (1970), 457โ479. MR 267586, DOI 10.1090/S0002-9947-1970-0267586-7
- M. G. Barratt and P. J. Hilton, On join operations in homotopy groups, Proc. London Math. Soc. (3) 3 (1953), 430โ445. MR 60232, DOI 10.1112/plms/s3-3.1.430
- Ralph L. Cohen and Paul Goerss, Secondary cohomology operations that detect homotopy classes, Topology 23 (1984), no.ย 2, 177โ194. MR 744849, DOI 10.1016/0040-9383(84)90038-7
- F. R. Cohen and M. E. Mahowald, Unstable properties of $\Omega ^{n}S^{n+k}$, Symposium on Algebraic Topology in honor of Josรฉ Adem (Oaxtepec, 1981), Contemp. Math., vol. 12, Amer. Math. Soc., Providence, R.I., 1982, pp.ย 81โ90. MR 676319 โ, The splitting of ${\Omega ^2}{S^{2n + 1}}$, Algebraic Topology (Seattle, 1985), Lecture Notes in Math., vol. 1286, Springer-Verlag 1987, pp. 188-192.
- F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. (2) 109 (1979), no.ย 1, 121โ168. MR 519355, DOI 10.2307/1971269
- F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. (2) 110 (1979), no.ย 3, 549โ565. MR 554384, DOI 10.2307/1971238
- F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Exponents in homotopy theory, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp.ย 3โ34. MR 921471
- F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Decompositions of loop spaces and applications to exponents, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp.ย 1โ12. MR 561210
- Brayton Gray, On the homotopy groups of mapping cones, Proc. London Math. Soc. (3) 26 (1973), 497โ520. MR 334198, DOI 10.1112/plms/s3-26.3.497
- Brayton Gray, Unstable families related to the image of $J$, Math. Proc. Cambridge Philos. Soc. 96 (1984), no.ย 1, 95โ113. MR 743705, DOI 10.1017/S0305004100061971
- Brayton Gray, On Todaโs fibrations, Math. Proc. Cambridge Philos. Soc. 97 (1985), no.ย 2, 289โ298. MR 771822, DOI 10.1017/S0305004100062836
- Brayton Gray, On the double suspension, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp.ย 150โ162. MR 1000375, DOI 10.1007/BFb0085226
- Brayton Gray, On the iterated suspension, Topology 27 (1988), no.ย 3, 301โ310. MR 963632, DOI 10.1016/0040-9383(88)90011-0 โ, Homotopy commutativity and the $EHP$ sequence, Proc. Internat. Conf., 1988, Contemp. Math., vol. 96, Amer. Math. Soc., Providence, R.I., 1989, pp. 181-188. โ, $EHP$ spectra and periodicity. II: $\Lambda$ algebra models, Trans. Amer. Math. Soc.
- John R. Harper, A proof of Grayโs conjecture, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp.ย 189โ195. MR 1022681, DOI 10.1090/conm/096/1022681 P. O. Kirley, On the indecomposability of iterated loop spaces, Ph.D. Thesis, Northwestern Univ., 1975.
- Joseph Neisendorfer, Properties of certain $H$-spaces, Quart. J. Math. Oxford Ser. (2) 34 (1983), no.ย 134, 201โ209. MR 698206, DOI 10.1093/qmath/34.2.201 D. C. Ravenel, private communication.
- Larry Smith, On realizing complex bordism modules. IV. Applications to the stable homotopy groups of spheres, Amer. J. Math. 99 (1977), no.ย 2, 418โ436. MR 433450, DOI 10.2307/2373828
- V. P. Snaith, A stable decomposition of $\Omega ^{n}S^{n}X$, J. London Math. Soc. (2) 7 (1974), 577โ583. MR 339155, DOI 10.1112/jlms/s2-7.4.577
- Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
- Hirosi Toda, An important relation in homotopy groups of spheres, Proc. Japan Acad. 43 (1967), 839โ842. MR 230310
- Hirosi Toda, On spectra realizing exterior parts of the Steenrod algebra, Topology 10 (1971), 53โ65. MR 271933, DOI 10.1016/0040-9383(71)90017-6
Bibliographic Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 340 (1993), 595-616
- MSC: Primary 55P40; Secondary 55P42, 55Q52
- DOI: https://doi.org/10.1090/S0002-9947-1993-1152323-X
- MathSciNet review: 1152323