TITLE: Computation of homotopy groups of spheres via Pontryagin-Thom
QUESTION [18 upvotes]: The Pontryagin-Thom construction identifies $\pi_{n+k}(S^n)$ with the group of bordism classes of framed $k$-dimensional submanifolds of $S^n$.  Before Serre's work introduced algebraic tools into the subject, this was used to calculate $\pi_{n+k}(S^n)$ for $0 \leq k \leq 2$ by Pontryagin and for $k=3$ by Rokhlin.  
Does there exist a modern exposition of these proofs anywhere?  The case $k=0$ is trivial, but as far as I can tell, the only sources for $k \geq 1$ are Pontryagin's book "Smooth manifolds and their application to homotopy theory" and Rochlin's original paper.  The book is very old-fashioned and spend way too much time developing the foundations of smooth manifold theory (I guess there was no nice source in the early 1950's), and Rokhlin's paper is unreadable (to me).

REPLY [8 votes]: I agree with you that Pontryagin's book is very hard to read!  And while there are many places that have brief sketches of what is going on (some given in the other answers), I am not aware of any other serious exposition of its contents.  To correct this, I have written a detailed modern account of Pontryagin's approach to calculating $\pi_{n+1}(S^n)$ and $\pi_{n+2}(S^n)$  in my notes "Homotopy groups of spheres and low-dimensional topology", which are available on my page of notes here.  I did not include a discussion of Rochlin's theorem, which is quite a bit harder.  The book of "A la recherche de la topologie Perdue" by Guillou and Marin (mentioned in Mike-Doherty's answer) is a good place to start for that.