Cobordism Obstructions to Complex Sections II: Torsion Obstructions

TL;DR

This paper investigates torsion obstructions to the existence of complex sections on almost complex manifolds, extending previous rational results by analyzing torsion phenomena using spectral sequences and showing vanishing results for certain orders.

Contribution

It introduces the study of torsion obstructions in complex sections, providing calculations with spectral sequences and establishing vanishing results for specific torsion orders.

Findings

Torsion obstructions vanish for low r in certain cases.

Spectral sequence calculations elucidate torsion obstructions.

Torsion obstructions for order p vanish when r<p^2-p for prime p≥3.

Abstract

In the previous paper, we studied obstructions to the existence of complex sections on almost complex manifolds up to cobordism. We determined the obstruction rationally, in terms of the Chern classes. In this paper, we study the torsion obstructions, that is, the obstructions which vanish after tensoring with $\mathbb{Q}$ or multiplication by an integer. Calculations with the Adams-Novikov spectral sequence for the Thom spectra $\mathbf{MTU}(d)$ allow us to show the torsion obstructions for low $r$. For prime $p\geq 3$, we show that torsion obstructions for finding $r$ complex sections of order $p$ vanish for $r<p^2-p$.