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 (� = 0) black hole spacetimes, and Gibbons’ and Woolgar’s genus-dependent, lower entropy bound for topological black holes in asymptotically locally antide Sitter ( �< 0) spacetimes. In higher dimensions the genus is replaced by the so-called σ -constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.