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In this paper we explore the topological classes of thermodynamics of afamily of black holes. In particular we investigate the influence of distinctfields, including the electric field, non-linear magnetic field, along with theperfect fluid matter background that can mimic dark matter in large distances.In light of these considerations, we shall henceforth denote this fluid asperfect fluid dark matter (PFDM). Our analysis reveals that the winding andtopological numbers for the Schwarzschild and Kerr black holes in PFDMbackground are the same as the Schwarzschild and Kerr black holes, however forthe Kerr-AdS background in PFDM we obtain a different topological numbercompared to the Kerr black hole in PFDM. Furthermore, we explore in details theinterplay of electric charge and nonlinear magnetic charge, impacting thetopological classes of thermodynamics both in the absence and presence of PFDM.Interestingly, it is demonstrated that the topological numbers associated withthe static Hayward black holes, both in the absence and presence of PFDM,deviate from those of the Schwarzschild black hole. This observation shows thatthe presence of a magnetic charge introduces an additional role and can alterthe topological numbers. Finally, our study culminates with the comprehensiveanalysis of the topological numbers pertaining to the Hayward black hole,considering the combined effects of PFDM and rotation.

Topological classes of thermodynamics of black holes in perfect fluid dark matter background