Characteristic rank of a real vector bundle
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Contents |
1 Definition


















Bundle dependency then admits the following definition of the upper characteristic rank of -complex
.






2 Motivation
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].
3 Examples
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.
4 References
- [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