# Galloway & Schoen's generalization of the black hole topology theorem

Abstract: In 1972, Hawking proved the black hole topology theorem, which states that a black hole horizon in a `classical setting' (i.e. a 4-dimensional asymptotically flat spacetime obeying the dominant energy condition) must be homeomorphic to $\mathbb{S}^{2}$. However, in 2002, Emparan and Reall came up with an example of a black hole horizon homeomorphic to $\mathbb{S}^{2}\times\mathbb{S}^{1}$ in a 5-dimensional spacetime. In this talk I will discuss the 2006 result of Galloway and Schoen that generalizes Hawking's result to higher dimensions: it states that in a spacetime satisfying the dominant energy condition, black hole horizons admit metrics of positive scalar curvature, which imposes fundamental restrictions on the topology of these horizons. This expository talk should be accessible to anyone who has taken the manifolds sequence or has some undergraduate experience with Riemannian geometry.