Abstract
A new 2-local spectrum Y is constructed so that H*Y is a cyclic A-module which in degrees ≤ 23 is the quotient of the Steenrod algebra by the left ideal generated by Sq 1, Sq 2, and Sq 4. In order to show that in this range Y splits off MO〈8〉, the groups π i (MO〈8〉) are calculated for i < 23. This includes a novel Adams differential. When i ≥ 16, these are the cobordism groups of 7-connected manifolds.
A sketch of the applicability of Y to obstruction theory is given.
1980 Mathematics subject classifications
- 57R42
- 57R90
- 55P42
Both authors were supported by National Science Foundation research grants
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
A. P. Bahri and M. Mahowald, A direct summand in H*MO〈8〉, Proc. Amer. Math. Soc. 78 (1980), 295–298.
D. M. Davis, Connective coverings of BO and immersions of projective spaces, Pac. Jour. Math. 76 (1978), 33–42.
_____, Some new immersions and nonimmersions of real projective spaces, Contemp. Math. 19 (1983), 33–42.
_____, A strong nonimmersion theorem for real projective spaces, Annals of Math 120 (1984), 517–528.
D. M. Davis, S. Gitler, W. Iberkleid, and M. Mahowald, The orientability of vector bundles with respect to certain spectra, Bol. Soc. Mat. Mex 268 (1981), 49–55.
D. M. Davis, S. Gitler, and M. Mahowald, The stable geometric dimension of vector bundles over real projective spaces, Trans. Amer. Math. Soc. 268 (1981), 39–62.
D. M. Davis and M. Mahowald, The geometric dimension of some vector bundles over projective spaces, Trans. Amer. Math. Soc. 205 (1975), 295–316.
_____, The immersion conjecture for RP 8ℓ+7 is false, Trans. Amer. Math. Soc. 236 (1978), 361–383.
_____, v 1-and v 2-periodicity in stable homotopy theory, Amer. Jour. Math. 103 (1978), 615–659.
_____, The nonrealizability of the quotient A//A 2 of the Steenrod algebra, Amer. Jour. Math. 104 (1982), 1211–1216.
_____, Ext over the subalgebra A 2 of the Steenrod algebra for stunted real projective spaces, Current Trends in Algebraic Topology, Conference, Proc. of Canadian Math. Soc. 2 (1982), 297–343.
V. Giambalvo, On 〈8〉 cobordism, Ill. Jour. Math. 15 (1971), 533–541.
_____, Correction to [12], Ill. Jour. Math. 16 (1972), p. 704.
S. Gitler and M. Mahowald, The geometric dimension of real stable vector bundles, Bol. Soc. Mat. Mex. 11 (1966), 85–107.
P. Goerss, J. Jones and M. Mahowald, Some generalized Brown-Gitler spectra, Trans Amer. Math. Soc. 294 (1986), 113–132.
M. W. Hirsch, Immersion of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276.
M. Mahowald, The metastable homtopy of S n, Mem. Amer. Math. Soc. 72 (1967).
_____, A new infinite family in 2π8, Topology 16 (1977), 249–256.
_____, Ring spectra which are Thom complexes, Duke Math. Jour. 46 (1979), 549–559.
_____, bo-resolutions, Pac. Jour. Math. 92 (1981), 365–383.
M. Mahowald and R. Rigdon, Obstruction theory with coefficients in a spectrum, Trans. Amer. Math. Soc. 204 (1975), 365–385.
M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349–369.
J. Milnor, A procedure for killing homotopy groups of differentiable manifolds, Proc. Symp. Pure Math. 3 (1961), 39–55.
M. Tangora, On the cohomology of the Steenrod algebra, Math. Zeit 116 (1970), 18–64.
H. Toda, “Composition methods in the homotopy groups of spheres,” Ann. of Math. Studies vol 45, Princeton Univ. Press, 1962.
C. T. C. Wall, “Surgery on compact manifolds,” Academic Press, 1970.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this paper
Cite this paper
Davis, D.M., Mahowald, M. (1989). A new spectrum related to 7-connected cobordism. In: Carlsson, G., Cohen, R., Miller, H., Ravenel, D. (eds) Algebraic Topology. Lecture Notes in Mathematics, vol 1370. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085223
Download citation
DOI: https://doi.org/10.1007/BFb0085223
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51118-2
Online ISBN: 978-3-540-46160-9
eBook Packages: Springer Book Archive
