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Motivated by the work of A. C. Naolekar and A. S. Thakur (2014) we introduce notions of upper chern rank and even cup length of a finite connected CW-complex and prove that upper chern rank is a homotopy invariant. It turns out that determination of upper chern rank of a space sometimes helps to detect whether a generator of the top cohomology group can be realized as Euler class for some real (orientable) vector bundle over or not. For a closed connected -dimensional complex manifold we obtain...
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