  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
($\Lambda=0$) black hole spacetimes, and Gibbons' and Woolgar's genus
dependent, lower entropy bound for topological black holes in asymptotically
locally anti-de Sitter ($\Lambda<0$) 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.
