The role of the equation of state and the space-time dimension in spherical collapse

We study the spherically symmetric collapse of a fluid with non-vanishing radial pressure in higher dimensional space-time. We obtain the general exact solution in the closed form for the equation of state ($P_r=\gamma \rho$) which leads to the explicit construction of the root equation governing the nature (black hole versus naked singularity) of the central singularity. A remarkable feature of the root equation is its invariance for the three cases: ($D+1,\gamma =-1$), ($D,\gamma =0$) and ($D-1,\gamma =1$) where $D$ is the dimension of space-time. That is, for the ultimate end result of the collapse, $D$-dimensional dust, $D+1$ - AdS (anti de Sitter)-like and $D-1$ - dS-like are absolutely equivalent.