Cobordism Obstructions to Complex Sections

Abstract

There is a classical problem to determine whether a manifold admits $r$ linearly independent tangent vector fields. The first results, due to Poincare and Hopf, show that an oriented manifold admits an everywhere non-zero vector field if and only if its Euler characteristic is zero. Thomas, Mayer, Atiyah and Dupont did further work showing the existence of obstructions whose vanishing was a necessary condition for a manifold to admit $r$ vector fields. To solve this problem up to cobordism, B\"okstedt and Svane defined and studied a notion of vector field cobordism for oriented manifolds and a corresponding cobordism category. The main goal of this thesis is to study the complex version of this problem, namely finding linearly independent complex tangent sections of almost complex manifolds. We define the complex section cobordism groups and the related cobordism categories. We identify an obstruction to finding a manifold in the same complex cobordism class as a given manifold with $r$ complex sections. This obstruction is an element of a relevant bordism group. The vanishing of this obstruction is both necessary and sufficient to show a cobordism class contains a manifold which can be equipped with $r$ linearly independent complex sections. Up to torsion, we completely describe this obstruction in terms of the Chern characteristic numbers. Further, calculations with the Adams-Novikov spectral sequence for particular Thom spectra allow us to show the torsion in the obstruction vanishes for low values of $r$. For prime $p\geq 3$, we show that torsion obstructions of order $p$ for finding $r$ complex sections vanish for $r<p^2-p$ and that all torsion obstructions for finding $2$ or $3$ linearly independent complex sections vanish. Finally, we show that this obstruction vanishes for certain multiplicative generators in the complex cobordism ring.