The -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

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

MathSciNet review:
876468

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Due to its relation to the image of the -homomorphism and first order periodicity (Bott periodicity), connective real -theory is well suited for problems in -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 -terms. In this paper we make a first attempt to construct an algebraic and computational theory of the -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 -term for in 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.

**[1]**J. F. Adams,*Stable homotopy and generalized homology*, University Press, Chicago, 1974. MR**0402720 (53:6534)****[2]**D. Davis, S. Gitler and M. Mahowald,*The stable geometric dimension of vector bundles over real projective spaces*, Trans. Amer. Math. Soc.**268**(1981), 39-61. MR**628445 (83c:55006)****[3]**-, Correction to*The stable geometric dimension of vector bundles over real projective spaces*, Trans. Amer. Math. Soc.**280**(1983), 841-843. MR**716854 (85g:55006)****[4]**J. Goldman and G.-C. Rota,*On the foundations of combinatorial theory*IV;*Finite vectorspaces and Eulerian generating functions*, Stud. Appl. Math.**49**(1970), 239-258. MR**0265181 (42:93)****[5]**W. Lellmann,*Operations and cooperations in odd-primary connective**-theory*, J. London Math. Soc. (2)**29**(1984), 562-576. MR**754942 (88a:55020)****[6]**-,*The**-term of the Adams Spectral sequence based on connected Morava**-theory*, preprint.**[7]**M. Mahowald,*Ring spectra which are Thom complexes*, Duke Math. J.**46**(1979), 549-559. MR**544245 (81f:55010)****[8]**-,*bo resolutions*, Pacific J. Math.**92**(1981), 365-383. MR**618072 (82m:55017)****[9]**-,*An addendum to bo-resolutions*, Pacific J. Math.**111**(1984), 117-123. MR**732062 (86a:55018)****[10]**-,*The image of**in the EHP-sequence*, Ann. of Math. (2)**116**(1982), 65-112. MR**662118 (83i:55019)****[11]**M. Mahowald and R. J. Milgram,*Operations which detect**in connective*theory, Quart. J. Math. Oxford Ser. (2)**28**(1976), 415-432. MR**0433453 (55:6429)****[12]**H. R. Margolis,*Eilenberg-Mac Lane spectra*, Proc. Amer. Math. Soc.**43**(1974), 409-415. MR**0341488 (49:6239)****[13]**R. J. Milgram,*The Steenrod algebra and its dual for connective**-theory*, Reunion sobre homotopia, Northwestern Univ., 1974 (Sociedad Matematica Mexicana, 1975), pp. 127-158. MR**761725****[14]**H. R. Miller,*On relations between Adams spectral sequences with an application to the stable homotopy of the Moore space*, J. Pure Appl. Algebra**20**(1981), 187-312. MR**604321 (82f:55029)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
55T15,
55N15,
55Q45,
55S25

Retrieve articles in all journals with MSC: 55T15, 55N15, 55Q45, 55S25

Additional Information

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

Keywords:
Connective -theory,
Adams spectral sequences,
stable homotopy groups,
-theory operations,
filtration spectral sequences

Article copyright:
© Copyright 1987
American Mathematical Society