Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Vicenç Navarro, Vicenç Aznar[en] The first Poincaré conjecture asserts that homology is sufficient to tell if a $n$-manifold is homeomorphic to a hypersphere. We describe a counterexample, named Poincaré homology sphere. Finnaly, we introduce new hypothesis to enunciate the last Poincaré conjecture.