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

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.

References

J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0402720
Donald M. Davis, Sam Gitler, and Mark Mahowald, The stable geometric dimension of vector bundles over real projective spaces, Trans. Amer. Math. Soc. 268 (1981), no. 1, 39–61. MR 628445, DOI 10.1090/S0002-9947-1981-0628445-8
Donald M. Davis, Sam Gitler, and Mark Mahowald, Correction to: “The stable geometric dimension of vector bundles over real projective spaces” [Trans. Amer. Math. Soc. 268 (1981), no. 1, 39–61; MR0628445 (83c:55006)], Trans. Amer. Math. Soc. 280 (1983), no. 2, 841–843. MR 716854, DOI 10.1090/S0002-9947-1983-0716854-X
Jay Goldman and Gian-Carlo Rota, On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions, Studies in Appl. Math. 49 (1970), 239–258. MR 265181, DOI 10.1002/sapm1970493239
Wolfgang Lellmann, Operations and co-operations in odd-primary connective $K$-theory, J. London Math. Soc. (2) 29 (1984), no. 3, 562–576. MR 754942, DOI 10.1112/jlms/s2-29.3.562

The

-term of the Adams Spectral sequence based on connected Morava

-theory

Mark Mahowald, Ring spectra which are Thom complexes, Duke Math. J. 46 (1979), no. 3, 549–559. MR 544245
Mark Mahowald, $b\textrm {o}$-resolutions, Pacific J. Math. 92 (1981), no. 2, 365–383. MR 618072
Mark Mahowald, An addendum to: “$b\textrm {o}$-resolutions” [Pacific J. Math. 92 (1981), no. 2, 365–383; MR0618072 (82m:55017)], Pacific J. Math. 111 (1984), no. 1, 117–123. MR 732062
Mark Mahowald, The image of $J$ in the $EHP$ sequence, Ann. of Math. (2) 116 (1982), no. 1, 65–112. MR 662118, DOI 10.2307/2007048
M. Mahowald and R. James Milgram, Operations which detect Sq4 in connective $K$-theory and their applications, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 108, 415–432. MR 433453, DOI 10.1093/qmath/27.4.415
H. R. Margolis, Eilenberg-Mac Lane spectra, Proc. Amer. Math. Soc. 43 (1974), 409–415. MR 341488, DOI 10.1090/S0002-9939-1974-0341488-9
R. James Milgram, The Steenrod algebra and its dual for connective $K$-theory, Conference on homotopy theory (Evanston, Ill., 1974) Notas Mat. Simpos., vol. 1, Soc. Mat. Mexicana, México, 1975, pp. 127–158. MR 761725
Haynes R. Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Algebra 20 (1981), no. 3, 287–312. MR 604321, DOI 10.1016/0022-4049(81)90064-5