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Transactions of the American Mathematical Society

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

 
 

 

The geometric dimension of some vector bundles over projective spaces


Authors: Donald M. Davis and Mark E. Mahowald
Journal: Trans. Amer. Math. Soc. 205 (1975), 295-315
MSC: Primary 55F25; Secondary 57D20
DOI: https://doi.org/10.1090/S0002-9947-1975-0372854-9
MathSciNet review: 0372854
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Abstract: We prove that in many cases the geometric dimension of the $ p$-fold Whitney sum $ p{H_k}$ of the Hopf bundle $ {H_k}$ over quaternionic projective space $ Q{P^k}$ is the smallest $ n$ such that for all $ i \leq k$ the reduction of the $ i$th symplectic Pontryagin class of $ p{H_k}$ to coefficients $ {\pi _{4i - 1}}(({\text{R}}{P^\infty }/{\text{R}}{P^{n - 1}})\Lambda bo)$ is zero, where bo is the spectrum for connective KO-theory localized at 2. We immediately obtain new immersions of real projective space $ {\text{R}}{P^{4k + 3}}$ in Euclidean space if the number of 1's in the binary expansion of $ k$ is between 5 and 8.


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  • [1] J. F. Adams, Quillen's work on formal groups and complex cobordism, mimeographed lecture notes, Chicago, 1970.
  • [2] D. W. Anderson, The real $ K$-theory of classifying spaces, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 634-636.
  • [3] D. M. Davis, Generalized homology and the generalized vector field problem, Quart. J. Math. Oxford Ser. (2) 25 (1974), 169-193. MR 0356053 (50:8524)
  • [4] S. Gitler, Immersion and embedding of manifolds, Proc. Sympos. Pure Math., vol. 22, Amer. Math. Soc., Providence, R. I., 1971, pp. 87-96. MR 47 #4275. MR 0315726 (47:4275)
  • [5] S. Gitler and M. E. Mahowald, Obstruction theory and $ K$-theory (mimeograph).
  • [6] -, The geometric dimension of real stable vector bundles, Bol. Soc. Mat. Mexicana (2) 11 (1966), 85-107. MR 37 #6922. MR 0231367 (37:6922)
  • [7] S. Gitler, M. E. Mahowald and R. J. Milgram, The nonimmersion problem for $ {\text{R}}{P^n}$ and higher-order cohomology operations, Proc. Nat. Acad. Sci. U. S. A. 60 (1968), 432-437. MR 37 #3581. MR 0227997 (37:3581)
  • [8] D. H. Husemoller, Fibre bundles, McGraw-Hill, New York, 1966. MR 37 #4821. MR 0229247 (37:4821)
  • [9] I. M. James, Spaces associated with Stiefel manifolds, Proc. London Math. Soc. (3) 9 (1959), 115-140. MR 21 #1596. MR 0102810 (21:1596)
  • [10] L. L. Larmore, Twisted cohomology theories and the single obstruction to lifting, Pacific J. Math. 41 (1972), 755-769. MR 0353315 (50:5799)
  • [11] M. E. Mahowald, The metastable homotopy of $ {S^n}$, Mem. Amer. Math. Soc. No. 72 (1967). MR 38 #5216.
  • [12] R. J. Milgram, Immersing projective spaces, Ann. of Math. (2) 85 (1967), 473-482. MR 35 #2293. MR 0211412 (35:2293)
  • [13] J. C. Moore, Some applications of homology theory to homotopy problems, Ann. of Math. (2) 58 (1953), 325-350. MR 15, 549. MR 0059549 (15:549a)
  • [14] D. Randall, Note on the generalized vector field problem, Bol. Soc. Mat. Mexicana (2) 17 (1972), 40-41. MR 0334235 (48:12554)
  • [15] B. J. Sanderson, Immersions and embeddings of projective spaces, Proc. London Math. Soc. (3) 14 (1964), 137-153. MR 29 #2814. MR 0165532 (29:2814)
  • [16] E. H. Spanier, Function spaces and duality, Ann. of Math. (2) 70 (1959), 338-378. MR 21 #6584. MR 0107862 (21:6584)
  • [17] G. W. Whitehead, Generalized homology theories, Trans. Amer. Math. Soc. 102 (1962), 227-283. MR 25 #573. MR 0137117 (25:573)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0372854-9
Keywords: Geometric dimension, immersions of projective spaces, symplectic Pontryagin classes, connective $ K$-theory, modified Postnikov towers
Article copyright: © Copyright 1975 American Mathematical Society

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