Pacific Journal of Mathematics

Vector bundles over $(8k+3)$-dimensional manifolds.

Tze Beng Ng

Article information

Source
Pacific J. Math., Volume 121, Number 2 (1986), 427-443.

Dates
First available in Project Euclid: 8 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.pjm/1102702442

Mathematical Reviews number (MathSciNet)
MR819199

Zentralblatt MATH identifier
0574.57010

Subjects
Primary: 57R25: Vector fields, frame fields
Secondary: 55S45: Postnikov systems, $k$-invariants

Citation

Ng, Tze Beng. Vector bundles over $(8k+3)$-dimensional manifolds. Pacific J. Math. 121 (1986), no. 2, 427--443. https://projecteuclid.org/euclid.pjm/1102702442


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References

  • [6] Applications. 6.1. Let M = S+Sk X P2/ + 1for k > 0, / > 0. Then we have THEOREM. San(M) > 8. Proof. Plainly Hn~5{M) Hn~6(M) = 0, w4(M) = 0 and 2(M) =
  • [11] M immersesin R2""7. Consequently we have by Theorem 5.8: THEOREM 6.4. SupposeM is 4-connected mod 2 and dim M = n = 11 m o d l 6 > l l .Then M immerses in R2n~7if Sq2 Hn~\M\Z) = Sq2 H"-(M) and immerses inR2n~s if Sq2Sq H"-\M)= Sq2Hn~\M). Similarly we have by Theorem 5.7: THEOREM 6.5. Suppose M satisfies conditions A and B 6>/3,w4(M) = 0 and n = ll mod 16 > 11. // Sq1 H\M)= Sq2 H\M)= 0 and if either Sq4 Hn~\M)= 0 or 3H\M) = 0 and Sq2 Hn\M)= 0 /e f /m- merses in R2"~8.
  • [I] J. Adem and S. Gitler, Secondary characteristicclasses and the immersion problem, Bol. Soc. Mat. Mexicana, 8 (1963), 53-78.
  • [2] S. Gitler and M. E. Mahowald, The geometric dimension of real stable vector bundles, Bol. Soc. Mat. Mexicana, 8 (1960), 85-106.
  • [3] A. Hughes and E. Thomas, A note on certain secondary cohomology operations, Bol. Soc. Mat. Mexicana, 71 (1970), 661-667.
  • [4] M. E. Mahowald, Theindex of a tangent 2-field, Pacific J. Math., 58 (1975), 539-548.
  • [5] M. E. Mahowald and F. P. Peterson, Secondarycohomology operations on the Thorn class, Topology, 2 (1964), 367-377.
  • [6] C. R. F. Maunder, Cohomology operations of the N-th kind, Proc. London Math. Soc, (3) (1960), 125-154.
  • [7] J. Milgram, Cartan Formulae, Illinois J. Math.,75 (1971), 633-647.
  • [8] Tze-Beng Ng, The existence of 1-fields and 8-fields on manifolds, Quart. J. Math. Oxford, (2) 30 (1979), 197-221.
  • [9] Tze-Beng Ng, The mod 2 cohomology ofBSOn(l6), to appear in Canad. J. Math.
  • [10] D. G. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann., 194 (1971), 197-212.
  • [II] D. Randall, Tangentframe fields on spin manifolds,Pacific J. Math., 76 (1978), 157-167.
  • [12] E. Thomas, Postniko invariants and higher order cohomology operations, Ann. of Math., (2) 85 (1967), 184-217.
  • [13] E. Thomas, Real and complex vector fields on manifolds, J. Math, and Mechanic, 16 (1967), 1183-1205.
  • [14] E. Thomas, The index of a tangent 2-fields,Comment. Math. Helv., 42 (1967), 86-110.
  • [15] E. Thomas, The span of a manifold, Quart. J. Math., Oxford (2) 19 (1968), 225-244.