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On Dold-Whitney's parallelizability of 4-manifolds

Valentina Bais

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.

On Dold-Whitney's parallelizability of 4-manifolds

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 -bundles over the 4-sphere by means of their Euler and first Pontryagin classes as a main tool.

Paper Structure

This paper contains 1 section, 2 theorems, 25 equations.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

A closed orientable 4-manifold $M$ is parallelizable if and only if its second Stiefel-Whitney class $w_2(M)$, first Pontryagin class $p_1(M)$ and Euler characteristic $\chi(M)$ are all vanishing.

Theorems & Definitions (4)

  • Theorem 1: Dold-Whitney
  • Lemma 2
  • Remark 3
  • Remark 4