# Cobordism Obstructions to Complex Sections

# Cobordism Obstructions to Complex Sections

There is a classical problem to determine whether a manifold admits r linearly independent tangent vector fields. In the case of one everywhere non-zero vector field, this problem was solved by Hopf, and the obstruction is the Euler characteristic of the manifold. Bokstedt, Dupont and Svane approached this problem by instead determining the obstruction to finding a cobordant manifold with r vector fields.

We extend their results by looking at obstructions to finding linearly independent complex sections of the tangent bundle of almost complex manifolds. In this case, we are able to describe the rational obstruction for almost complex manifolds. This obstruction is given in terms of Chern characteristic numbers. Moreover, we are able to give certain bounds for r under which the torsion obstruction vanishes.