# Computation of homotopy groups of spheres via Pontryagin-Thom

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).

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I still believe that Pontryagin's book is by far the best presentation of the subject. Unlike more modern books, Pontryagin makes a systematic effort to describe the key ideas unobscured by technicalities. At the same time no technical detail is missing. This is some achievement! Even the foundational part is a joy to read. You should compare Pontryagin's presentation of the proof of Sard's theorem to Milnor's. The proofs are identical, but for the first time I saw somebody who could outdo Milnor in presenting the proof. –  Liviu Nicolaescu Nov 19 '12 at 10:12
Naturally Pontryagin's book does not contain the computation of the third stable homotopy group of spheres, which is Rokhlin's result, but it does the computation of the first two groups. –  András Szűcs Oct 6 '13 at 12:53
The too much time Pontryagin spent on developing foundation could be related to the fact that the first published version of the computation of the second stable homotopy group contained a mistake and that led to an incorrect result for it. That was corrected more than ten years later. –  András Szűcs Oct 6 '13 at 13:03
Pontryagin's mistake was, that he thought that the Arf invariant was a homomorphism instead of a quadratic function (that is $q(x+y) \ne q(x) + q(y)$, but $q(x+y) = q(x) + q(y) + xy).$ –  András Szűcs Oct 6 '13 at 13:17

In "Lecture notes in algebraic topology" by J. David and P. Kirk, they explain in chapter $8$, the Thom-Pontryagin construction and how to relate bordism of stably framed manifolds to stable homotopy groups of spheres. There is also "Bordism, stable homotopy and Adams spectral sequences" by S. Kochman where he explains the cases $k=0,1$ in the first chapter. For the case $k=2$, I really like the first sections of the paper "Quadratic functions in geometry, topology and M-theory" by M. Hopkins and I. Singer. Or you can listen to M. Hopkins' lecture on the Kervaire invariant: http://empg.maths.ed.ac.uk/Videos/Atiyah80/Hopkins.mov the lecture is so cool and he spends some time explaining the computation of $\pi_2^S$. Slides are available here: http://www.maths.ed.ac.uk/~aar/atiyah80.htm

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The $k=0$ and $k=1$ case are drawn up nicely in the second appendix of Freed and Uhlenbeck's classic book Instantons and Four-Manifolds. It's entitled the Pontrjagin-Thom Construction, and is motivated by wanting to compute $[M,S^3]$ (for any compact simply-connected 4-manifold) whose nontriviality depends on the parity of the natural intersection form.
The best part: there is a cool picture of a dinosaur (that is, a framed cobordism) being cut open (that is, by two homotopy-equivalent framings).

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The first part of the book "A la recherche de la topologie Perdue" edited by Guillou and Marin (Progress in Mathematics no 62, Birkhauser 1986) has translations into French of Rokhlin's papers, followed by commentary which aims to assist in reading the papers, and to demonstrate the assertions of Rokhlin which don't seem obvious.

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Here the paper: A. Szűcs: Two theorems of Rokhlin. This gives the computation of $\pi(3)$ by elementary tools.

But it uses the following non-trivial fact: The 3-dimensional spin cobordism group is trivial.

You can read a proof of this latter in the same issue in the paper by A. Stipsicz.

But a more simple proof for this is in:

Melvin, P.; Kazez, W. 3-dimensional bordism. Michigan Math. J. 36 (1989), no. 2, 251–260.

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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.

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