On parallelizability and span of the Dold manifolds
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Abstract:
The Dold manifold $P(m,n)$ is obtained from the product $S^m \times \mathbb CP^n$ of the $m$-dimensional sphere and $n$-dimensional complex projective space by identifying $(x,[z_1, \dots , z_{n+1}])$ with $(-x,[\bar z_1, \dots , \bar z_{n+1}])$, where $\bar z$ denotes the complex conjugate of $z$. We answer the parallelizability question for the Dold manifolds $P(m,n)$ and, by completing an earlier (2008) result due to Peter Novotný, we solve the vector field problem for the manifolds $P(m,1)$.References
- J. F. Adams, Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962), 39–41. MR 133837, DOI 10.1090/S0002-9904-1962-10693-4
- R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87–89. MR 102804, DOI 10.1090/S0002-9904-1958-10166-4
- Donald Davis, Generalized homology and the generalized vector field problem, Quart. J. Math. Oxford Ser. (2) 25 (1974), 169–193. MR 356053, DOI 10.1093/qmath/25.1.169
- Albrecht Dold, Erzeugende der Thomschen Algebra ${\mathfrak {N}}$, Math. Z. 65 (1956), 25–35 (German). MR 79269, DOI 10.1007/BF01473868
- Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
- Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR 1249482, DOI 10.1007/978-1-4757-2261-1
- Július Korbaš, Distributions, vector distributions, and immersions of manifolds in Euclidean spaces, Handbook of global analysis, Elsevier Sci. B. V., Amsterdam, 2008, pp. 665–724, 1214. MR 2389644, DOI 10.1016/B978-044452833-9.50014-0
- Július Korbaš and Peter Zvengrowski, The vector field problem: a survey with emphasis on specific manifolds, Exposition. Math. 12 (1994), no. 1, 3–20. MR 1267626
- Ulrich Koschorke, Vector fields and other vector bundle morphisms—a singularity approach, Lecture Notes in Mathematics, vol. 847, Springer, Berlin, 1981. MR 611333
- Kee Yuen Lam, Sectioning vector bundles over real projective spaces, Quart. J. Math. Oxford Ser. (2) 23 (1972), 97–106. MR 296965, DOI 10.1093/qmath/23.1.97
- Kee Yuen Lam and Duane Randall, Geometric dimension of bundles on real projective spaces, Homotopy theory and its applications (Cocoyoc, 1993) Contemp. Math., vol. 188, Amer. Math. Soc., Providence, RI, 1995, pp. 137–160. MR 1349135, DOI 10.1090/conm/188/02239
- John Milnor, On the immersion of $n$-manifolds in $(n+1)$-space, Comment. Math. Helv. 30 (1956), 275–284. MR 79268, DOI 10.1007/BF02564347
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
- Himadri Kumar Mukerjee, Classification of homotopy Dold manifolds, New York J. Math. 9 (2003), 271–293. MR 2016190
- Peter Novotný, Span of Dold manifolds, Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 4, 687–698. MR 2475492
- R. E. Stong, Vector bundles over Dold manifolds, Fund. Math. 169 (2001), no. 1, 85–95. MR 1852354, DOI 10.4064/fm169-1-3
- Emery Thomas, Vector fields on manifolds, Bull. Amer. Math. Soc. 75 (1969), 643–683. MR 242189, DOI 10.1090/S0002-9904-1969-12240-8
- J. J. Ucci, Immersions and embeddings of Dold manifolds, Topology 4 (1965), 283–293. MR 187250, DOI 10.1016/0040-9383(65)90012-1
Additional Information
- Július Korbaš
- Affiliation: Department of Algebra, Geometry, and Mathematical Education, Faculty of Mathematics, Physics, and Informatics, Comenius University, Mlynská dolina, SK-842 48 Bratislava 4, Slovakia — and — Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava 1, Slovakia
- Email: korbas@fmph.uniba.sk
- Received by editor(s): November 10, 2011
- Published electronically: April 30, 2013
- Additional Notes: Part of this research was carried out while the author was a member of two research teams supported in part by the grant agency VEGA (Slovakia)
- Communicated by: Daniel Ruberman
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 2933-2939
- MSC (2010): Primary 57R25; Secondary 55S40, 57R20
- DOI: https://doi.org/10.1090/S0002-9939-2013-11573-X
- MathSciNet review: 3056583