On root invariants of periodic classes in 𝐸𝑥𝑡_{𝐴}(𝑍/2,𝑍/2)

Shick, Paul

doi:10.1090/S0002-9947-1987-0879570-3

On root invariants of periodic classes in $\textrm {Ext}_ A(\textbf {Z}/2,\textbf {Z}/2)$
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Trans. Amer. Math. Soc. 301 (1987), 227-237 Request permission

Abstract:

We prove that if a class in the cohomology of the mod 2 Steenrod algebra is $\operatorname {mod} 2$-periodic in the sense of [10], then its root invariant must be ${\upsilon _{n + 1}}$-periodic, where ${\upsilon _{n}}$ denotes the $n$th generator of ${\pi _ \ast }({\text {BP}})$.

References

Donald M. Davis and Mark Mahowald, Ext over the subalgebra $A_{2}$ of the Steenrod algebra for stunted projective spaces, Current trends in algebraic topology, Part 1 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 1982, pp. 297–342. MR 686123, DOI 10.2307/2374058
Donald M. Davis and Mark Mahowald, $v_{1}$- and $v_{2}$-periodicity in stable homotopy theory, Amer. J. Math. 103 (1981), no. 4, 615–659. MR 623131, DOI 10.2307/2374044
David Copeland Johnson and Zen-ichi Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka Math. J. 17 (1980), no. 1, 117–136. MR 558323
Wen Hsiung Lin, On conjectures of Mahowald, Segal and Sullivan, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 3, 449–458. MR 556925, DOI 10.1017/S0305004100056887
Wên Hsiung Lin, Algebraic Kahn-Priddy theorem, Pacific J. Math. 96 (1981), no. 2, 435–455. MR 637982, DOI 10.2140/pjm.1981.96.435
Wen Hsiung Lin, Cohomology of sub-Hopf-algebras of the Steenrod algebra, J. Pure Appl. Algebra 10 (1977/78), no. 2, 101–113. MR 454975, DOI 10.1016/0022-4049(77)90013-5
W. H. Lin, D. M. Davis, M. E. Mahowald, and J. F. Adams, Calculation of Lin’s Ext groups, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 3, 459–469. MR 569195, DOI 10.1017/S0305004100056899
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
Mark Mahowald and Paul Shick, Some root invariants in $\textrm {Ext}_{A}^{\ast } ,^{\ast } (\textbf {Z}/2,\,\textbf {Z}/2)$, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, R.I., 1983, pp. 227–231. MR 711054
Mark Mahowald and Paul Shick, Periodic phenomena in the classical Adams spectral sequence, Trans. Amer. Math. Soc. 300 (1987), no. 1, 191–206. MR 871672, DOI 10.1090/S0002-9947-1987-0871672-0
Haynes R. Miller, Douglas C. Ravenel, and W. Stephen Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (2) 106 (1977), no. 3, 469–516. MR 458423, DOI 10.2307/1971064
Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042
Douglas C. Ravenel, The Segal conjecture for cyclic groups and its consequences, Amer. J. Math. 106 (1984), no. 2, 415–446. With an appendix by Haynes R. Miller. MR 737779, DOI 10.2307/2374309
Reinhard Schultz, Transformation groups and exotic spheres, Group actions on manifolds (Boulder, Colo., 1983) Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 243–267. MR 780966, DOI 10.1090/conm/036/780966
Stephan Stolz, A note on conjugation involutions on homotopy complex projective spaces, Japan. J. Math. (N.S.) 12 (1986), no. 1, 69–73. MR 914308, DOI 10.4099/math1924.12.69