Black Hole Topology in $f(R)$ Gravity

Hawking’s topology theorem in general relativity restricts the cross-section of the event horizon of a black hole in $3+1$ dimension to be either spherical or toroidal. The toroidal case is ruled out by the topology censorship theorems. In this article, we discuss the generalization of this result to black holes in $f(R)$ gravity in $3+1$ and higher dimensions. We obtain a sufficient differential condition on the function $f^{\mathrm{\prime }}(R)$, which restricts the topology of the horizon cross-section of a black hole in $f(R)$ gravity in $3+1$ dimension to be either $S^2$ or $S^1\times S^1$. We also extend the result to higher dimensional black holes and show that the same sufficient condition also restricts the sign of the Yamabe invariant of the horizon cross-section.