El fibrado de Hopf en homotopía estable

Abstract

[en] In this work we study the Hopf map from the 3-sphere $S^{3} $ to the 2-sphere $S^{2}$. We review some properties of the higher homotopy groups of spaces and prove that the Hopf map is a generator of $\pi_{3} (S^{2})$.

As an introduction to stable homotopy theory, we prove the Freudenthal suspension theorem for the spheres and explain why the first stable homotopy group $\pi^{s}_{1}$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$. In order to prove it we use the Pontryagin-Thom construction, a result that relates the homotopy groups of spheres with framed cobordism classes of framed manifolds. Our goal is to understand geometrically why the class represented by the Hopf map has infinite order in $\pi_{3}(S^{2})$ but its suspensions have order 2 in $\pi_{n+1}(S^{n})$ for $n > 2$.