We give an upper bound for the characteristic rank of smooth closed null-cobordant manifolds and apply it to the Grassmann manifolds of oriented vector subspaces in Euclidean space. The bound turns out to be sharp: it coincides with the exact value of characteristic rank for the Grassmann manifolds of oriented 2-dimensional vector subspaces in odd-dimensional Euclidean space.
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M. Audin, Lagrangian skeletons, periodic geodesic flows and symplectic cuttings, Manuscripta Math., 124 (2007), 533–550.
A. Borel, La cohomologie mod 2 de certains espaces homogènes, Comment. Math. Helv., 27 (1953), 165–197.
W. Greub, S. Halperin and R. Vanstone, Connections, Curvature, and Cohomology, Vol. III – Cohomology of Principal Bundles and Homogeneous Spaces, Academic Press (New York, 1976).
L’. Horanská and J., Korbaš, On cup products in some manifolds, Bull. Belg. Math. Soc. – Simon Stevin, 7 (2000), 21–28.
J. Korbaš, On fibrations with Grassmannian fibers, Bull. Belg. Math. Soc. – Simon Stevin, 8 (2001), 119–130.
J. Korbaš, The cup-length of the oriented Grassmannians vs a new bound for zero-cobordant manifolds, Bull. Belg. Math. Soc. – Simon Stevin, 17 (2010), 69–81.
J. Milnor and J. Stasheff, Characteristic Classes, Ann. Math. Stud. 76, Princeton Univ. Press (Princeton, N.J., 1974).
A. Naolekar and A. Thakur, Note on the characteristic rank of vector bundles, Preprint, Indian Statistical Institute, Bangalore, India; http://www.isibang.ac.in/~statmath/eprints/2012/1.pdf, January 20, 2012.
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Balko, L., Korbaš, J. A note on the characteristic rank of null-cobordant manifolds. Acta Math Hung 140, 145–150 (2013). https://doi.org/10.1007/s10474-012-0279-3
Key words and phrases
- null-cobordant (cobordant to zero) manifold
- Stiefel–Whitney characteristic class
- characteristic rank
- oriented Grassmann manifold
Mathematics Subject Classification