Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Axial maps with further structure


Author: A. J. Berrick
Journal: Proc. Amer. Math. Soc. 54 (1976), 413-416
DOI: https://doi.org/10.1090/S0002-9939-1976-0397750-9
MathSciNet review: 0397750
Full-text PDF

Abstract | References | Additional Information

Abstract: For $ F = {\mathbf{R}},{\mathbf{C}}$ or $ {\mathbf{H}}$ an $ F$-axial map is defined to be an axial map $ {\mathbf{R}}{P^m} \times {\mathbf{R}}{P^m} \to {\mathbf{R}}{P^{m + k}}$ equivariant with respect to diagonal and trivial $ {F^{\ast}}$-actions. Analogously to the real case, it is shown that $ {\mathbf{C}}$-axial maps correspond to immersions of $ {\mathbf{C}}{P^n}$ in $ {{\mathbf{R}}^{2n + k}}$ while (for $ F = {\mathbf{R}}$ and for $ F = {\mathbf{C}}$, $ k$ odd) embeddings induce $ F$-symmaxial maps. Examples are thereby given of symmaxial maps not induced by embeddings of $ {\mathbf{R}}{P^n}$, and of $ {\mathbf{R}}$-axial maps which are not $ {\mathbf{C}}$-axial. Furthermore, the relationships which hold when $ F = {\mathbf{R}},{\mathbf{C}}$ are no longer valid for $ F = {\mathbf{H}}$.


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

  • [1] J. Adem and S. Gitler, Secondary characteristic classes and the immersion problem, Bol. Soc. Mat. Mexicana (2) 8 (1963), 53-78. MR 29 #5255. MR 0167990 (29:5255)
  • [2] J. Adem, S. Gitler and I. M. James, On axial maps of a certain type, Bol. Soc. Mat. Mexicana (2) 17 (1972), 59-62. MR 0336757 (49:1530)
  • [3] M. F. Atiyah, Thom complexes, Proc. London Math. Soc. (3) 11 (1961), 291-310. MR 24 #A1727 MR 0131880 (24:A1727)
  • [4] S. Gitler and M. Mahowald, Some immersions of real projective spaces, Bol. Soc. Mat. Mexicana (2) 14 (1969), 9-21. MR41 #2696. MR 0258049 (41:2696)
  • [5] 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)
  • [6] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242-276. MR 22 #9980. MR 0119214 (22:9980)
  • [7] D. Husemoller, Fibre bundles, McGraw-Hill, New York, 1966. MR 37 #4821. MR 0229247 (37:4821)
  • [8] I. M. James, Some embeddings of projective spaces, Proc. Cambridge Philos. Soc. 55 (1959), 294-298. MR 22 #236. MR 0109350 (22:236)
  • [9] J. Levine, Embedding and immersion of real projective spaces, Proc. Amer. Math. Soc. 14 (1963), 801-803. MR 27 #5272. MR 0155338 (27:5272)
  • [10] M. Mahowald, On the embeddability of the real projective spaces, Proc. Amer. Math. Soc. 13 (1962), 763-764. MR 26 #782. MR 0143222 (26:782)
  • [11] B. J. Sanderson, A non-immersion theorem for real projective space, Topology 2 (1963), 209-211. MR 27 #1968. MR 0151987 (27:1968)
  • [12] -, Immersions and embeddings of projective spaces, Proc. London Math. Soc. (3) 14 (1964), 137-153, MR 29 #2814. MR 0165532 (29:2814)
  • [13] 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)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0397750-9
Keywords: Axial map, embedding, immersion, projective space, skew map, symmaxial map, tangent bundle
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society