Vector fields on $RP^m\times RP^n$
HTML articles powered by AMS MathViewer
- by Donald M. Davis PDF
- Proc. Amer. Math. Soc. 140 (2012), 4381-4388 Request permission
Abstract:
The span of a manifold is its maximum number of linearly independent vector fields. We discuss the question, still unresolved, of whether $\operatorname {span}(P^m\times P^n)$ always equals $\operatorname {span}(P^m)+\operatorname {span}(P^n)$. Here $P^n$ denotes real projective space. We use $BP$-cohomology to obtain new upper bounds for $\operatorname {span}(P^m\times P^n)$, much stronger than previously known bounds.References
- J. C. Becker, The span of spherical space forms, Amer. J. Math. 94 (1972), 991β1026. MR 312516, DOI 10.2307/2373562
- Donald Davis, Generalized homology and the generalized vector field problem, Quart. J. Math. Oxford Ser. (2) 25 (1974), 169β193. MR 356053, DOI 10.1093/qmath/25.1.169
- Ioan James and Emery Thomas, An approach to the enumeration problem for non-stable vector bundles, J. Math. Mech. 14 (1965), 485β506. MR 0175134
- David Copeland Johnson, W. Stephen Wilson, and Dung Yung Yan, Brown-Peterson homology of elementary $p$-groups. II, Topology Appl. 59 (1994), no.Β 2, 117β136. MR 1296028, DOI 10.1016/0166-8641(94)90090-6
- Teiichi Kobayashi, Note on $\gamma$-dimension and products of real projective spaces, J. Math. Soc. Japan 34 (1982), no.Β 3, 501β505. MR 659618, DOI 10.2969/jmsj/03430501
- Hyun-Jong Song and W. Stephen Wilson, On the nonimmersion of products of real projective spaces, Trans. Amer. Math. Soc. 318 (1990), no.Β 1, 327β334. MR 979967, DOI 10.1090/S0002-9947-1990-0979967-7
- Haruo Suzuki, Operations in $KO$-theory and products of real projective spaces, Mem. Fac. Sci. Kyushu Univ. Ser. A 18 (1964), 140β153. MR 176489, DOI 10.2206/kyushumfs.18.140
Additional Information
- Donald M. Davis
- Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015
- MR Author ID: 55085
- Email: dmd1@lehigh.edu
- Received by editor(s): December 16, 2010
- Received by editor(s) in revised form: May 31, 2011
- Published electronically: April 18, 2012
- Communicated by: Brooke Shipley
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 4381-4388
- MSC (2010): Primary 57R25, 55N20
- DOI: https://doi.org/10.1090/S0002-9939-2012-11282-1
- MathSciNet review: 2957228