On the topology and area of higher dimensional black holes

Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higher dimensional analogues of some well known results for black holes in 3+1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking’s black hole topology theorem for asymptotically flat ($\mathrm{\Lambda }=0$) black hole spacetimes, and Gibbons’ and Woolgar’s genus dependent, lower entropy bound for topological black holes in asymptotically locally anti-de Sitter ($\mathrm{\Lambda }<0$) spacetimes. In higher dimensions the genus is replaced by the so-called $\sigma$-constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.