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)

 
 

 

Nonimmersion of lens spaces with 2-torsion


Author: A. J. Berrick
Journal: Trans. Amer. Math. Soc. 224 (1976), 399-405
MSC: Primary 57D40
DOI: https://doi.org/10.1090/S0002-9947-1976-0420662-3
MathSciNet review: 0420662
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: From a study of the equivariant unitary K-theory of the Stiefel manifold $ {V_{k + 1,2}}({\mathbf{C}})$, it is shown that the lens space $ {L^k}(n)$, with n a multiple of $ {2^{2k - 1 - \alpha (k - 1)}}$, does not immerse in Euclidean space of dimension $ 4k - 2\alpha (k) - 2$.


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

  • [1] M. F. Atiyah, K-theory, Benjamin, New York, 1967. MR 36 #7130. MR 0224083 (36:7130)
  • [2] A. J. Berrick, Obstructions in K-theory (to appear).
  • [3] A. J. Berrick, S. Feder and S. Gitler, Symmetric axial maps and embeddings of projective spaces, Bol. Soc. Mat. Mexicana (to appear).
  • [4] D. M. Davis and M. Mahowald, A strong nonimmersion theorem for $ R{P^{8l + 7}}$, Bull. Amer. Math. Soc. 81 (1975), 155-156. MR 0405420 (53:9213)
  • [5] S. Gitler, Immersion and embedding of manifolds, Algebraic Topology, Proc. Sympos. Pure Math., vol. 22, Amer. Math. Soc., Providence, R. I., 1971, pp. 87-96. MR 47 #4275. MR 0315726 (47:4275)
  • [6] A. Haefliger and M. W. Hirsch, Immersions in the stable range, Ann. of Math. (2) 75 (1962), 231-241. MR 26 #784. MR 0143224 (26:784)
  • [7] S. G. Hoggar, On KO theory of Grassmannians, Quart. J. Math. Oxford Ser. (2) 20 (1969), 447-463. MR 40 #8048. MR 0254841 (40:8048)
  • [8] I. M. James, On the immersion problem for real projective spaces, Bull. Amer. Math. Soc. 69 (1963), 231-238. MR 26 #1900. MR 0144355 (26:1900)
  • [9] T. Kambe, The structure of $ {K_\Lambda }$-rings of the lens space and their applications, J. Math. Soc. Japan 18 (1966), 135-146. MR 33 #6646. MR 0198491 (33:6646)
  • [10] T. Kobayashi, Non-immersion theorems for lens spaces, J. Math. Kyoto Univ. 6 (1966), 91-108. MR 36 #3371. MR 0220305 (36:3371)
  • [11] B. J. Sanderson and R. L. E. Schwarzenberger, Non-immersion theorems for differentiable manifolds, Proc. Cambridge Philos. Soc. 59 (1963), 319-322. MR 26 #5589. MR 0148080 (26:5589)
  • [12] D. Sjerve, Geometric dimension of vector bundles over lens spaces, Trans. Amer. Math. Soc. 134 (1968), 545-557. MR 38 #1695. MR 0233373 (38:1695)
  • [13] -, Vector bundles over orbit manifolds, Trans. Amer. Math. Soc. 138 (1969), 97-106. MR 38 #6621. MR 0238345 (38:6621)
  • [14] C. B. Thomas, Structures on manifolds defined by cross-sections, Math. Ann. 196 (1972), 163-170. MR 45 #9355. MR 0300309 (45:9355)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57D40

Retrieve articles in all journals with MSC: 57D40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0420662-3
Keywords: Complex G-vector bundle, complex Grassmannian, complex oriented Grassmannian, complex Stiefel manifold, equivariant unitary K-theory, immersion, lens space, projective tangent bundle
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society