  In the recent paper [Phys. Rev. Lett. 129, 191101 (2022)], the black holes
were viewed as topological thermodynamic defects by using the generalized
off-shell free energy. Their work indicates that all black hole solutions in
the pure Einstein-Maxwell gravity theory could be classified into three
different topological classes for four and higher spacetime dimensions. In this
paper, we investigate the topological number of BTZ black holes with different
charges $(Q)$ and rotational $(J)$ parameters. By using generalized free energy
and Duan's $\phi$-mapping topological current theory, we interestingly found
only two topological classes for BTZ spacetime. Particularly, for $Q=J=0$ BTZ
black hole, there has only one zero point and therefore the total topological
number is 1. While for rotating or charged cases, there are always two zero
points and the global topological number is zero.

General Relativity and Quantum Cosmology