Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T14:16:42.406Z Has data issue: false hasContentIssue false

Whitehead squares in Thom complexes

Published online by Cambridge University Press:  20 January 2009

W. A. Sutherland
Affiliation:
New College, Oxford, OX1 3BN
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

One of the aims of this paper is to examine the following conjecture, attributed to Mahowald on p. 255 of (2), Part 2. Let M be a closed connected smooth manifold of odd dimension q (q≠l,3,7) and with tangent bundle τ. Let the inclusion of a compactified fibre in the Thorn complex of τ be written μ: SqTτ.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1981

References

REFERENCES

(1)Adams, J. F., On the groups J(X)-I, Topology 2 (1963), 181–95.CrossRefGoogle Scholar
(2)A.M.S. Colloquia Proceedings, Vol. 32 (Providence, R.I., 1978).Google Scholar
(3)Atiyah, M. F., Thorn complexes, Proc. London Math. Soc. (3) 11 (1961), 291310.CrossRefGoogle Scholar
(4)Boardman, J. M. and Steer, B., On Hopf invariants, Comment. Math. Helv. 42 (1967), 180221.CrossRefGoogle Scholar
(5)Browder, W., The Kervaire invariant of framed manifolds and its generalizations, Ann. of Math. 90 (1969), 157–86.CrossRefGoogle Scholar
(6)Brown, E. H. Jnr., Generalizations of the Kervaire invariant, Ann. of Math. 95 (1972), 368–83.CrossRefGoogle Scholar
(7)Brown, E. H. Jnr and Peterson, F. P., Whitehead products and cohomology operations, Quart. J. Math. (Oxford) (2) 15 (1964), 116–20.CrossRefGoogle Scholar
(8)Dibag, I., Degree theory for spherical fibrations, to appear.Google Scholar
(9)Dupont, J. L., On homotopy invariance of the tangent bundle I, Math. Scand. 26 (1970), 513.CrossRefGoogle Scholar
(10)Dupont, J. L., On homotopy invariance of the tangent bundle II, Math. Scand. 26 (1970), 200220.CrossRefGoogle Scholar
(11)James, I. M. and Thomas, E., An approach to the enumeration problem for non-stable vector bundles, J. Math. Mech. 14 (1965), 485506.Google Scholar
(12)Marcum, H. J. and Randall, D., The homotopy Thom class of a spherical fibration, Proc. Amer. Math. Soc. 80 (1980), 353358.CrossRefGoogle Scholar
(13)Massey, W. S. and Peterson, F. P., On the dual Stiefel-Whitney classes of a manifold, Bol. Soc. Mat. Mexicana (2) 8 (1963), 113.Google Scholar
(14)Milgram, R. J. and Rees, Elmer, On the normal bundle to an embedding, Topology 10 (1971), 299308.CrossRefGoogle Scholar
(15)Sutherland, W. A., The fibre homotopy enumeration of non-stable sphere bundles and fibrings over real projective spaces, J. London Math. Soc. (2) 1 (1969), 693704.CrossRefGoogle Scholar
(16)Sutherland, W. A., The Browder-Dupont invariant, Proc. London Math. Soc. (3) 33 (1976), 94112.CrossRefGoogle Scholar
(17)Wall, C. T. C., Poincaré complexes: I, Ann. of Math. 86 (1967), 213245.CrossRefGoogle Scholar
(18)Whitehead, G. W., Elements of homotopy theory (Springer-Verlag, 1978).CrossRefGoogle Scholar