Cobordism Obstructions to Complex Sections II: Torsion Obstructions

In the previous paper, we studied the existence of complex sections on almost complex manifolds up to cobordism. We proved a necessary and sufficient condition for their existence in Ω*U⊗ Q in terms of the Chern classes. In this paper we study the existence question in ΩU_*. We call corresponding obstructions torsion and we determine them for small values of r. Calculations with the Adams-Novikov spectral sequence for the Thom spectra defined in the previous paper allow us to show the vanishing of the torsion obstruction for low r. For prime p, we show the vanishing of p torsion obstructions for finding r complex sections for r<p²-p.