Chern Rank of Complex Bundle

TL;DR

This paper introduces the concepts of upper chernrank and even cup length for CW-complexes, showing how upper chernrank can help identify Euler classes and providing bounds and methods for computing even cup length in specific cases.

Contribution

It defines new invariants like upper chernrank and explores their properties, including homotopy invariance and applications to complex manifolds and CW-complexes.

Findings

Upper chernrank is a homotopy invariant.

Determines bounds for even cup length in complex manifolds.

Provides methods to compute even cup length when upper chernrank equals dimension.

Abstract

We introduce notions of {\it upper chernrank} and {\it even cup length} of a finite connected CW-complex and prove that {\it upper chernrank} is a homotopy invariant. It turns out that determination of {\it upper chernrank} of a space $X$ 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 $X$ or not. For a closed connected $d$-dimensional complex manifold we obtain an upper bound of its even cup length. For a finite connected even dimensional CW-complex with its {\it upper chernrank} equal to its dimension, we provide a method of computing its even cup length. Finally, we compute {\it upper chernrank} of many interesting spaces.