On Dold-Whitney's parallelizability of 4-manifolds

Valentina Bais

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arXiv · 2410.22117

On Dold-Whitney's parallelizability of 4-manifolds

Valentina Bais

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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.

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Valentina Bais. 2024-10-29. On Dold-Whitney's parallelizability of 4-manifolds. https://arxiv.org/abs/2410.22117

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