In Dan Freed's notes, he asks us to compute $\pi_2(S^1), \pi_3(S^2), \pi_4(S^3)$ (See Exercise 1.50) using Pontryagin-Thom theorem. [Note: In the notes there is a typo. Freed refers to Hopf degree theorem for this exercise.]
How do we solve this exercise?
My thoughts: We have to classify framed 1-manifolds in $S^{n+1}$ upto cobordisms. Essentially we have to classify a collection of framed circles (upto cobordism).
The pair of pants bordism will show that a collection of framed circles (with same orientation) are cobordant to a single framed circle with a particular orientation.
By a bent cylinder, we can show that two framed circles with opposite orientations are cobordant to the empty manifold.
So to solve the problem I have to answer:
Are two framed circles cobordant?
I don't know how to answer this question and I am not sure if my reasoning is right.