On the double suspension homomorphism at odd primes

Authors:
J. R. Harper and H. R. Miller

Journal:
Trans. Amer. Math. Soc. **273** (1982), 319-331

MSC:
Primary 55T15; Secondary 55P40, 55Q25, 55Q45, 55U99

MathSciNet review:
664045

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Abstract: We work with the -term for spheres and the stable Moore space, given by the -algebra at odd primes. Writing and , we construct compatible maps and prove the Metastability Theorem: in homology induces an isomorphism for where stem degree, homological degree resulting from the bigrading of and . There is an operator corresponding to the Adams stable self-map of the Moore space and extends to . A corollary of the Metastability Theorem and the Localization Theorem of the second author is that the map induces an isomorphism on homology after inverting .

**[1]**J. F. Adams,*On the groups 𝐽(𝑋). IV*, Topology**5**(1966), 21–71. MR**0198470****[2]**A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector, and J. W. Schlesinger,*The 𝑚𝑜𝑑-𝑝 lower central series and the Adams spectral sequence*, Topology**5**(1966), 331–342. MR**0199862****[3]**A. K. Bousfield and D. M. Kan,*The homotopy spectral sequence of a space with coefficients in a ring.*, Topology**11**(1972), 79–106. MR**0283801****[4]**J. Harper,*Rank**-spaces*, Canad. Math. Soc. Conf. Proc., Current Trends in Algebraic Topology, Western Ontario, 1981.**[5]**D. C. Johnson, H. R. Miller, W. S. Wilson, and R. S. Zahler,*Boundary homomorphisms in the generalized Adams spectral sequence and the nontriviality of infinitely many 𝛾_{𝑡} in stable homotopy*, Conference on homotopy theory (Evanston, Ill., 1974) Notas Mat. Simpos., vol. 1, Soc. Mat. Mexicana, México, 1975, pp. 47–63. MR**761720****[6]**Arunas Liulevicius,*Zeroes of the cohomology of the Steenrod algebra*, Proc. Amer. Math. Soc.**14**(1963), 972–976. MR**0157383**, 10.1090/S0002-9939-1963-0157383-7**[7]**Mark Mahowald,*On the double suspension homomorphism*, Trans. Amer. Math. Soc.**214**(1975), 169–178. MR**0438333**, 10.1090/S0002-9947-1975-0438333-5**[8]**Haynes R. Miller,*A localization theorem in homological algebra*, Math. Proc. Cambridge Philos. Soc.**84**(1978), no. 1, 73–84. MR**0494105****[9]**Haynes Miller and Clarence Wilkerson,*Vanishing lines for modules over the Steenrod algebra*, J. Pure Appl. Algebra**22**(1981), no. 3, 293–307. MR**629336**, 10.1016/0022-4049(81)90104-3**[10]**Martin C. Tangora,*Some remarks on the lambda algebras*, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 476–487. MR**513587****[11]**F. R. Cohen,*The unstable decomposition of**and its applications*(to appear).

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DOI:
https://doi.org/10.1090/S0002-9947-1982-0664045-2

Article copyright:
© Copyright 1982
American Mathematical Society