Characteristic rank of a real vector bundle
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Bundle dependency then admits the following definition of the upper characteristic rank of -complex .
For closed, smooth, connected -dimensional manifold unorientedly cobordant to zero, there exist an element in -cohomology algebra of , which cannot be expressed as a polynomial in Stiefel-Whitney classes of (tangent bundle of) manifold . For this type of manifold there is the following estimate of cuplength of manifold .
Theorem 2.1.[Korbaš2010] Let be a closed smooth connected -dimensional manifold unorientedly cobordant to zero. Let , , be the first nonzero reduced cohomology group of . Let be an integer such that for each element of can be expressed as a polynomial in the Stiefel-Whitney classes of the manifold . Then we have that
This leads the definition of characteristic rank of manifold , which is the largest possible which fulfills the conditions of the theorem above. Making the definition of characteristic rank bundle dependent, i.e. replacing manifold with connected, finite -complex and tangent bundle of with a real vector bundle , gives definition 1.1 [Naolekar&Thakur2014].
The cohomology ring of -dimensional real projective space is , where is the first Stiefel-Whitney characteristic class of canonical line bundle over [Milnor&Stasheff1974]. It is then clear, that . On the other hand, we have for odd and for k even. In this case equals if is odd, and equals if is even.
- [Korbaš2010] 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.
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Naolekar&Thakur2014] A. C. Naolekar and A. S. Thakur, Note on the characteristic rank of vector bundles, Math. Slovaca (2014) 64: 1525. https://doi.org/10.2478/s12175-014-0289-4