2024/10/29 by Bais, Valentina
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2410.22117
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.