The 𝑏𝑜-Adams spectral sequence

Lellmann, Wolfgang; Mahowald, Mark

doi:10.1090/S0002-9947-1987-0876468-1

The $b\textrm {o}$-Adams spectral sequence
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by Wolfgang Lellmann and Mark Mahowald

Trans. Amer. Math. Soc. 300 (1987), 593-623

DOI: https://doi.org/10.1090/S0002-9947-1987-0876468-1

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Abstract:

Due to its relation to the image of the $J$-homomorphism and first order periodicity (Bott periodicity), connective real $K$-theory is well suited for problems in $2$-local stable homotopy that arise geometrically. On the other hand the use of generalized homology theories in the construction of Adams type spectral sequences has proved to be quite fruitful provided one is able to get a hold on the respective ${E_2}$-terms. In this paper we make a first attempt to construct an algebraic and computational theory of the ${E_2}$-term of the bo-Adams spectral sequence. This allows for some concrete computations which are then used to give a proof of the bounded torsion theorem of [8] as used in the geometric application of [2]. The final table of the ${E_2}$-term for $\pi _ \ast ^S$ in ${\operatorname {dim}} \leq 20$ shows that the statement of this theorem cannot be improved. No higher differentials appear in this range of the bo-Adams spectral sequence. We observe, however, that such a differential has to exist in dim 30.

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The

-term of the Adams Spectral sequence based on connected Morava

-theory

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