Is (S^n) X R Parallelizable for All n?

[amd939]

Hi, I am new to manifold and having a hard time on it. Could anyone please help me on the following problem. Please write down your thoughts. Thanks alot.

Prove that (S^n) X R is parallelizable for all n.

 

[shoehorn]

Show us what you've done so far and tell us precisely where you're stuck.

 

[amd939]

Hi,I tried to show S^nXR is parallelizable by showing S^nXR is diffeomorphic to R^n+1\{0} so if R^n+1\{0} is parallelizable then the problem solved. But I just don't know how to show R^n+1\{0} is parallelizable.

 

[Reverie]

Hi,I tried to show S^nXR is parallelizable by showing S^nXR is diffeomorphic to R^n+1\{0} so if R^n+1\{0} is parallelizable then the problem solved. But I just don't know how to show R^n+1\{0} is parallelizable.

R^(n+1)\{0} is a subset of R^(n+1). You can explicitly write down a global trivialization for R^(n+1) that restricts to a global frame for R^(n+1)\{0}.

A global frame for R^(n+1) is...

(1,0,...,0)
(0,1,0,...,0)
.
.
.
(0,...,0,1)