Is the tangent bundle of $S^2 \times S^1$ trivial or not?



Tangent bundle of $S^2 \times S^1$ trivial or not
It is trivial.


More generally, by a result of Kervaire a product of any number $n \geq 2$ of spheres is a parallelizable manifold if one of them has odd dimension.


For a short proof see 


*E. B. Staples*, [**A short and elementary proof that a product of spheres is parallelizable if one of them is odd**](http://www.ams.org/mathscinet-getitem?mr=219082), *Proc. Amer. Math. Soc.* **18** (1967), 570--571.


