Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T13:39:09.043Z Has data issue: false hasContentIssue false
  • English
  • Français

Codimension one immersions and the Kervaire invariant one problem

Published online by Cambridge University Press:  24 October 2008

Peter John Eccles
Peter John Eccles
Affiliation:
University of Manchester
Get access Rights & Permissions [Opens in a new window]

Extract

Let i: M↬ℝn+1 be a self-transverse immersion of a compact closed smooth n-dimensional manifold in (n + 1)-dimensional Euclidean space. A point of ℝn+1 is an r-fold intersection point of the immersion if it is the image under i of (at least) r distinct points of the manifold. The self-transversality of i implies that the set of r-fold intersection points is the image of an immersion of a manifold of dimension n+1-r (the empty set if r > n + 1). In particular, the set of (n + l)-fold intersection points is finite of order, say, θ(i). In this paper we are concerned with the set of values of θ(i) for (self-transverse) immersions of all (compact closed smooth) manifolds of given dimension n.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 3 , November 1981 , pp. 483 - 493
Copyright
Copyright © Cambridge Philosophical Society 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Adams, J. F.On the non-existence of elements of Hopf invariant one. Ann. of Math. (2) 72 (1960), 20–104.Google Scholar
(2)Banchoff, T. F.Triple points and surgery of immersed surfaces. Proc. Amer. Math. Soc. 46 (1974), 407–413.CrossRefGoogle Scholar
(3)Barratt, M. G. and Eccles, P. J.Γ+-structures III: The stable structure of Ω∞Σ∞A. Topology 13 (1974), 199–207.CrossRefGoogle Scholar
(4)Barratt, M. G. and Mahowald, M. E. Unpublished.Google Scholar
(5)Boardman, J. M. and Steer, B.On Hopf invariants. Comment. Math. Helv. 42 (1967), 180–221.Google Scholar
(6)Browder, W.Homology operations and loop spaces. Illinois. J. Math. 4 (1960), 347–357.Google Scholar
(7)Browder, W.The Kervaire invariant of framed manifolds and its generalization. Ann. of Math. (2) 90 (1969), 157–186.CrossRefGoogle Scholar
(8)Cohen, F. R., Lada, T. J. and May, J. P.The homology of iterated loop spaces. Lecture Notes in Math. no. 533 (Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
(9)Dyer, E. and Lashof, R. K.Homology of iterated loop spaces. Amer. J. Math. 84 (1962), 35–88.CrossRefGoogle Scholar
(10)Eccles, P. J. Multiple points of codimension one immersions. Topology Symposium Siegen 1979, pp. 23–38. Lecture Notes in Math. no. 788 (Springer-Verlag, Berlin, Heidelberg, New York, 1980).Google Scholar
(11)Eccles, P. J.Multiple points of codimension one immersions of oriented manifolds. Math. Proc. Cambridge Philos. Soc. 87 (1980), 213–220.CrossRefGoogle Scholar
(12)Hirsch, M. W.Immersions of manifolds. Trans. Amer. Math. Soc. 93 (1959), 242–276.CrossRefGoogle Scholar
(13)James, I. M.On the suspension triad. Ann. of Math. (2) 63 (1956), 191–247.Google Scholar
(14)Kahn, D. S. Homology of the Barratt-Eccles decomposition maps. Conference on Homotopy Theory Evanston 1974, pp. 65–82. Notas de Matemática y Simposia no. 1 (Soc. Mat. Mexicana, Mexico, 1975).Google Scholar
(15)Kahn, D. S. and Priddy, S. B.The transfer and stable homotopy theory. Math. Proc. Cambridge Philos. Soc. 83 (1978), 103–111.CrossRefGoogle Scholar
(16)Koschorke, U.Multiple points of immersions, and the Kahn-Priddy theorem. Math. Z. 169 (1979), 223–236.CrossRefGoogle Scholar
(17)Koschorke, U. and Sanderson, B. J.Self-intersections and higher Hopf invariants. Topology 17 (1978), 283–290.CrossRefGoogle Scholar
(18)Lannes, J. Immersions et homomorphisme d'Hurewicz (to appear).Google Scholar
(19)Mahowald, M. E. and Tangora, M. C.Some differentials in the Adams spectral sequence. Topology 6 (1967), 349–369.CrossRefGoogle Scholar
(20)Milnor, J. and Moore, J. C.On the structure of Hopf algebras. Ann. of Math. (2) 81 (1965), 211–264.CrossRefGoogle Scholar
(21)Nishida, G.The nilpotency of elements of the stable homotopy groups of spheres. J. Math. Soc. Japan 25 (1973), 707–732.Google Scholar
(22)Peterson, F. P. and Stein, N.Secondary cohomology operations: two formulas. Amer. J. Math. 81 (1959), 281–305.CrossRefGoogle Scholar
(23)Snaith, V. P.A stable decomposition of ΩnSnX. J. London Math. Soc. (2) 7 (1974), 577–583.CrossRefGoogle Scholar
(24)Thom, R.Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28 (1954), 17–86.Google Scholar
(25)Wells, R.Cobordism groups of immersions. Topology 5 (1966), 281–294.CrossRefGoogle Scholar