Can $f(T)$ gravity resolve the $H_0$ tension?

Motivated by the discrepancy in measurements of $H_0$ between local and global probes, we investigate whether teleparallel gravities could be a better model to describe the present days observations or at least to alleviate the $H_0$ tension. Specifically, in this work we study and place constraints on three popular $f(T)$ models in light of the Planck-2018 CMB data release. We find that the $f(T)$ power-law model can alleviate the $H_0$ tension from $4.4\sigma$ to $1.9\sigma$ level, while the $f(T)$ model of two exponential fail to resolve this inconsistency. Moreover, for the first time, we obtain constraints on the effective number of relativistic species $N_{eff}$ and on the sum of the neutrino masses $\Sigma m_\nu$ in $f(T)$ gravity. We find that the constraints obtained are looser than in $\Lambda$CDM. However, the introduction of massive neutrinos into the cosmological model alleviate the $H_0$ tension for the power-law model. Finally, we find that whether a viable $f(T)$ theory can mitigate the $H_0$ tension depends on the mathematical structure of the distortion factor $y(z,\,b)$. These results could provide a clue for theoreticians to write a more physical-motivated expression of $f(T)$ function.