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A new spectrum related to 7-connected cobordism

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1370)

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

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References

  1. A. P. Bahri and M. Mahowald, A direct summand in H*MO〈8〉, Proc. Amer. Math. Soc. 78 (1980), 295–298.

    MathSciNet  MATH  Google Scholar 

  2. D. M. Davis, Connective coverings of BO and immersions of projective spaces, Pac. Jour. Math. 76 (1978), 33–42.

    CrossRef  MATH  Google Scholar 

  3. _____, Some new immersions and nonimmersions of real projective spaces, Contemp. Math. 19 (1983), 33–42.

    MathSciNet  Google Scholar 

  4. _____, A strong nonimmersion theorem for real projective spaces, Annals of Math 120 (1984), 517–528.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. 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.

    MathSciNet  MATH  Google Scholar 

  6. 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.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. D. M. Davis and M. Mahowald, The geometric dimension of some vector bundles over projective spaces, Trans. Amer. Math. Soc. 205 (1975), 295–316.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. _____, The immersion conjecture for RP 8ℓ+7 is false, Trans. Amer. Math. Soc. 236 (1978), 361–383.

    MathSciNet  MATH  Google Scholar 

  9. _____, v 1-and v 2-periodicity in stable homotopy theory, Amer. Jour. Math. 103 (1978), 615–659.

    CrossRef  MathSciNet  Google Scholar 

  10. _____, The nonrealizability of the quotient A//A 2 of the Steenrod algebra, Amer. Jour. Math. 104 (1982), 1211–1216.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. _____, 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.

    MathSciNet  MATH  Google Scholar 

  12. V. Giambalvo, On 〈8〉 cobordism, Ill. Jour. Math. 15 (1971), 533–541.

    MathSciNet  MATH  Google Scholar 

  13. _____, Correction to [12], Ill. Jour. Math. 16 (1972), p. 704.

    MathSciNet  MATH  Google Scholar 

  14. S. Gitler and M. Mahowald, The geometric dimension of real stable vector bundles, Bol. Soc. Mat. Mex. 11 (1966), 85–107.

    MathSciNet  MATH  Google Scholar 

  15. P. Goerss, J. Jones and M. Mahowald, Some generalized Brown-Gitler spectra, Trans Amer. Math. Soc. 294 (1986), 113–132.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. M. W. Hirsch, Immersion of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. M. Mahowald, The metastable homtopy of S n, Mem. Amer. Math. Soc. 72 (1967).

    Google Scholar 

  18. _____, A new infinite family in 2π8, Topology 16 (1977), 249–256.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. _____, Ring spectra which are Thom complexes, Duke Math. Jour. 46 (1979), 549–559.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. _____, bo-resolutions, Pac. Jour. Math. 92 (1981), 365–383.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. M. Mahowald and R. Rigdon, Obstruction theory with coefficients in a spectrum, Trans. Amer. Math. Soc. 204 (1975), 365–385.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349–369.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. J. Milnor, A procedure for killing homotopy groups of differentiable manifolds, Proc. Symp. Pure Math. 3 (1961), 39–55.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. M. Tangora, On the cohomology of the Steenrod algebra, Math. Zeit 116 (1970), 18–64.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. H. Toda, “Composition methods in the homotopy groups of spheres,” Ann. of Math. Studies vol 45, Princeton Univ. Press, 1962.

    Google Scholar 

  26. C. T. C. Wall, “Surgery on compact manifolds,” Academic Press, 1970.

    Google Scholar 

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© 1989 Springer-Verlag

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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

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  • DOI: https://doi.org/10.1007/BFb0085223

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51118-2

  • Online ISBN: 978-3-540-46160-9

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