Obstruction theory on 8-manifolds

TL;DR

This paper provides a comprehensive obstruction-theoretic framework for 8-manifolds, establishing cohomological criteria for complex and quaternionic structures and structure group reductions, advancing understanding of manifold geometry.

Contribution

It introduces a uniform, self-contained approach to obstruction problems on 8-manifolds, with explicit cohomological conditions for various geometric structures.

Findings

Necessary and sufficient cohomological criteria for almost complex structures

Criteria for almost quaternionic structures on tangent bundles

Conditions for reducing structure group to U(3)

Abstract

This note gives a uniform, self-contained, and fairly direct approach to a variety of obstruction-theoretic problems on 8-manifolds. We give necessary and sufficient cohomological critera for the existence of almost complex and almost quaternionic structures on the tangent bundle and for the reduction of the structure group to U(3) by the homomorphism U(3) --> O(8) given by the Lie algebra representation of PU(3).