  We study the property of matter in equilibrium with a static, spherically
symmetric black hole in D-dimensional spacetime. It requires this kind of
matter has an equation of state (\omega\equiv p_r/\rho=-1/(1+2kn), k,n\in
\mathbb{N}), which seems to be independent of D. However, when we associate
this with specific models, some interesting limits on space could be found:
(i)(D=2+2kn) while the black hole is surrounded by cosmic strings;
  (ii)the black hole can be surrounded by linear dilaton field only in
4-dimensional spacetime. In both cases, D=4 is special.
