Topología diferencial: grado de Brouwer-Kronecker

Abstract

In this work we study the Brouwer-Kronecker degree, using de Rham cohomology and integration of forms on manifolds. As first applications we deduce two important theorems due to Brouwer: the fixed point theorem and the so-called hairy ball theorem. But our main motivation are the homotopy groups of spheres πn(Sm). These groups are all trivial for n < m. This is easy and we deduce it using the Sard-Brown theorem. For n = m the Brouwer-Kronecker degree gives the solution: πm(Sm) = Z, a famous theorem by Hopf which is the central result we prove here. For n > m homotopy groups become much more complicated, except the case n = 1 whose elementary computation we include for completeness. Still, degree gives us a method to understand some cases n > m > 1 using the Hopf invariant. We define this invariant and obtain its basic properties. Using all of this we compute the Hopf invariant of the famous Hopf fibration, to conclude that the group π3(S2) is infinite.