Exotic smooths structures on spheres

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Ricardo García López [en] The main goal of this work is to prove the existence of exotic smooth spheres. These are smooth manifolds that are homeomorphic but not diffeomorphic to the standard sphere. This was first shown in the 7-dimensional case by John Milnor in his influential paper [Mil56], and this work replicates his construction. In order to state and prove this result, though, a journey through some background is needed. This includes singular homology and cohomology theory, Morse theory and characteristic classes. Hence, we also slightly develop these topics here.