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

Wolfgang Lellmann, Mark Mahowald

1987
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Transactions of the American Mathematical Society
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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
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... c and computational theory of the E2-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 wI in dim", 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. In this paper we analyze the E 1 -and E 2 -terms of the Adams spectral sequence based on real connective K-theory boo As can be seen from applications [2, 8, 10] and our sample calculations, this spectral sequence is quite powerful. It converges to the 2-local stable homotopy groups 'lTt(X)(2) ' Unfortunately its E 2 -term lacks computability due to the fact that an algebraic description is not yet known. In this paper we show that the E 2 -term can be embedded into a long exact sequence of which at least one of the other two terms does not have this disadvantage: in most interesting cases it can be described algebraically (as a certain Ext-functor) and it is completely computable in examples like spheres or stunted (real) projective spaces. Tables for X = SO suggest that in dimensions ~ 45 nearly all of the classes found in this way detect in fact homotopy classes. We now give a more detailed account of the contents of the individual chapters. In §1 we fix some notations and describe the algebra of operations in bo and bsp (up to torsion). This was done additively in [13] . The starting point of the whole analysis is the splitting of bo-module spectra 4n bol\bo"=' V LbOI\B(n), n;;.O where B( n) denotes an integral Brown-Gitler spectrum [8]. . Primary 55T15, 55N15. 55S25. 55Q45.

doi:10.1090/s0002-9947-1987-0876468-1
fatcat:ibegznh7zzcszomdqvvgpim2dy