User n zhao - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:55:33Z http://mathoverflow.net/feeds/user/29218 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/113801/computation-of-homotopy-groups-of-spheres-via-pontryagin-thom Computation of homotopy groups of spheres via Pontryagin-Thom N Zhao 2012-11-19T05:55:57Z 2012-11-19T19:49:34Z <p>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. </p> <p>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).</p>