Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The $ b{\rm o}$-Adams spectral sequence

Authors: Wolfgang Lellmann and Mark Mahowald
Journal: Trans. Amer. Math. Soc. 300 (1987), 593-623
MSC: Primary 55T15; Secondary 55N15, 55Q45, 55S25
MathSciNet review: 876468
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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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Keywords: Connective $ K$-theory, Adams spectral sequences, stable homotopy groups, $ K$-theory operations, filtration spectral sequences
Article copyright: © Copyright 1987 American Mathematical Society