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Odd primary periodic phenomena in the classical Adams spectral sequence


Author: Paul Shick
Journal: Trans. Amer. Math. Soc. 309 (1988), 77-86
MSC: Primary 55T15; Secondary 55Q45
DOI: https://doi.org/10.1090/S0002-9947-1988-0938921-2
MathSciNet review: 938921
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Abstract: We study certain periodic phenomena in the cohomology of the $ \bmod \;p$ Steenrod algebra which are related to the polynomial generators $ {v_n} \in {\pi _{\ast}}BP$. A chromatic resolution of the $ {E_2}$ term of the classical Adams spectral sequence is constructed.


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  • [1] J. F. Adams, On the groups $ J(X)$, IV, Topology 5 (1966), 21-71. MR 0198470 (33:6628)
  • [2] N. A. Baas, On bordism theory of manifolds with singularity, Math. Scand. 33 (1973), 279-302. MR 0346824 (49:11547b)
  • [3] M. G. Barratt, Homotopy operations and homotopy groups (mimeographed notes), Seattle, 1963.
  • [4] D. M. Davis and M. E. Mahowald, $ {v_1}$- and $ {v_2}$-periodicity in stable homotopy, Amer. J. Math. 103 (1981), 615-659. MR 623131 (82j:55017)
  • [5] -, $ \operatorname{Ext} $ over the subalgebra $ {A_2}$ for stunted projective spaces, Current Trends in Algebraic Topology, Canad. Math. Soc. Conf. Proc., vol. 2, part 1, 1982, pp. 297-342.
  • [6] E. Devinatz, M. J. Hopkins and J. Smith, Nilpotence in stable homotopy, Ann. of Math. (to appear).
  • [7] D. C. Johnson and Z.-I. Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka J. Math. 17 (1980), 117-136. MR 558323 (81b:55010)
  • [8] W.-H. Lin, D. M. Davis, M. E. Mahowald, and J. F. Adams, Calculation of Lin's $ \operatorname{Ext} $-groups, Math. Proc. Cambridge Philos. Soc. 87 (1980), 459-469. MR 569195 (81e:55025)
  • [9] M. E. Mahowald and P. L. Shick, Periodic phenomena in the classical Adams spectral sequence, Trans. Amer. Math. Soc. 300 (1987), 191-206. MR 871672 (88e:55019)
  • [10] -, Root invariants and periodicity in stable homotopy, Bull. London Math. Soc. 20 (1988), 262-266. MR 931189 (89b:55009)
  • [11] H. R. Miller, A localization theorem in homological algebra, Math. Proc. Cambridge Philos. Soc. 84 (1978), 73-84. MR 0494105 (58:13036)
  • [12] -, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Algebra 20 (1981), 287-312. MR 604321 (82f:55029)
  • [13] H. R. Miller, D. C. Ravenel and W. S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. 166 (1977), 469-516. MR 0458423 (56:16626)
  • [14] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, 1986. MR 860042 (87j:55003)
  • [15] -, The geometric realization of the chromatic resolution, Algebraic Topology and Algebraic $ K$-Theory, Ann. of Math. Studies, no. 113, Princeton Univ. Press, 1987, pp. 168-179. MR 921477 (89d:55050)
  • [16] P. L. Shick, On root invariants of periodic classes in $ {\operatorname{Ext} _A}({\mathbf{Z}}/2,\,{\mathbf{Z}}/2)$, Trans. Amer. Math. Soc. 301 (1987), 227-337. MR 879570 (88f:55027)
  • [17] L. Smith, On realizing complex bordism modules, IV, Amer. J. Math. 99 (1977), 418-436. MR 0433450 (55:6426)

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DOI: https://doi.org/10.1090/S0002-9947-1988-0938921-2
Article copyright: © Copyright 1988 American Mathematical Society

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