On Dold-Whitney's parallelizability of 4-manifolds

TL;DR

This paper proves that a closed orientable 4-manifold is parallelizable precisely when its key characteristic classes vanish, using a detailed argument based on the classification of SO(4)-bundles over the 4-sphere.

Contribution

It provides a detailed proof of Dold-Whitney's criterion for parallelizability of 4-manifolds, clarifying the role of characteristic classes and bundle classification.

Findings

Parallelizability characterized by vanishing characteristic classes

Utilizes classification of SO(4)-bundles over the 4-sphere

Connects Dold-Whitney's result with explicit bundle classification

Abstract

We present a proof of the fact that a closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class and Euler characteristics vanish. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. The contribution of this note is to outline in detail an argument which is essentially due to R. Kirby, using the classification of $SO(4)$-bundles over the 4-sphere by means of their Euler and first Pontryagin classes as a main tool.