Dan Gregorian Fodor: On the parallelizability of tangent bundles for $2$ and $3$-dimensional manifolds, 387-401

Abstract:

We provide necessary and sufficient conditions for the parallelizability of $TM$, where $M$ is a $2$ or $3$-manifold.

Given a $2$-manifold $M$, its tangent bundle $TM$ is paralelizable if and only if $M$ is noncompact or of even Euler characteristic. We give two proofs, one using Stiefel-Whitney and Wu classes, another using obstruction theory and the classification theorem for compact surfaces.

For a $3$-manifold $M$, its tangent bundle $TM$ is paralelizable if and only if the cup product of the first Stiefel-Whitney class of $M$ with itself is zero.

Key Words: Obstruction theory, tangent bundles, paralelizability, characteristic classes.

2010 Mathematics Subject Classification: Primary 57R22; Secondary 57R25, 55R40.