Homotopy Groups of Spheres using the Pontryagin-Thom Theorem

Abstract

In this thesis, we calculate the first and second stable homotopy groups of spheres, using a geometric approach. First we prove the Pontryagin theorem relating the homotopy groups of spheres to equivalence classes of manifolds. We classify the first two cases of this, thereby calculating the first two stable homotopy groups and quickly look at extending this approach further. At the end, we look at the history of this theorem and this approach to calculating these groups.