By Dimas Tejada and Ingrid Martínez (Universidad de El Salvador).
Starting at a basic level, for maps between topological spaces the compact and open topology will be analyzed as well as the exponential law. The concepts of smooth manifolds and maps will be introduced, and the main results on the Weak and Strong Topologies on the Space of C^r-Maps between manifolds will be described. Finally, we will introduce the degree modulo 2 and the Brouwer degree of smooth maps. The course will finish by discussing the celebrated Poincaré-Hopf Theorem.