Topological classes of rotating black holes

In this paper, we investigate the topological numbers for the singly rotating Kerr black holes in all dimensions and the four-dimensional Kerr-Newman black hole. We show that for uncharged black holes, the rotation parameter has a significant effect on the topological number, and for rotating black holes, the dimension of spacetimes has a remarkable effect on the topological number too. In addition, we find that the topological numbers of the four-dimensional Kerr and Kerr-Newman black holes are the same, which seems to indicate that the electric charge parameter has no effect on the topological number of rotating black holes. Our current research provides more evidence that the conjecture put forward in Wei et al. Phys. Rev. Lett., 129, 191101 (2022), according to which all black hole solutions should be separated into three different topological classes, is accurate, at least in the pure Einstein-Maxwell gravity theory.