Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Unstable bordism groups and isolated singularities


Author: David Ellis
Journal: Trans. Amer. Math. Soc. 274 (1982), 695-708
MSC: Primary 57R75
DOI: https://doi.org/10.1090/S0002-9947-1982-0675075-9
MathSciNet review: 675075
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An isolated singularity of an embedded submanifold can be topologically smoothed if and only if a certain obstruction element in $ {\pi _ \ast }(MG)$ vanishes, where $ G$ is the group of the normal bundle. In fact this obstruction lies in a certain subgroup which is referred to here as the unstable $ G$-bordism group. In this paper some of the unstable $ SO$-bordism groups are computed; the obstruction to smoothing the complex cone on an oriented submanifold $ X \subset \mathbf{C}{P^n}$ at $ \infty$ is computed in terms of the characteristic numbers of $ X$. Examples of nonsmoothable complex cone singularities are given using these computations.


References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, Stable homotopy and generalized homology, Univ. of Chicago Press, Chicago, 1974. MR 0402720 (53:6534)
  • [2] W. Browder, Torsion in $ H$-spaces, Ann. of Math. (2) 74 (1961), 24-52. MR 0124891 (23:A2201)
  • [3] O. Burlet, Cobordisms de plongements et produits homotopiques, Comment. Math. Helv. 46 (1971), 277-289. MR 0295367 (45:4433)
  • [4] D. Ellis, Unstable oriented bordism groups and some applications, Ph.D Thesis, University of California, Berkeley, 1981.
  • [5] A. Haefliger, Plongements différentiables de variétés dans variétés, Comment. Math. Helv. 36 (1961), 42-82. MR 0145538 (26:3069)
  • [6] W. Massey and F. P. Peterson, On the dual Stiefel-Whitney classes of a manifold. Bol. Soc. Mat. Mexicana (2) 8 (1963), 1-13. MR 0163325 (29:628)
  • [7] R. J. Milgram, Unstable homotopy from the stable point of view, Springer-Verlag, Berlin, Heidelberg and New York, 1974. MR 0348740 (50:1235)
  • [8] E. Rees and E. Thomas, Cobordism obstructions to deforming isolated singularities, Math. Ann. 232 (1978), 33-55. MR 0500994 (58:18475)
  • [9] G. W. Whitehead, On the homology suspension, Ann. of Math. (2) 62 (1955), 254-268. MR 0073989 (17:520b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57R75

Retrieve articles in all journals with MSC: 57R75


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0675075-9
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society