Exploiting the Persistent Homology technique and its complementary representations, 
we examine the footprint of summed neutrino mass (    ) in the various density fields simulated by the publicly available Quijote suite. The evolution of topological features by utilizing the super-level filtration on three-dimensional density fields at zero redshift, reveals a remarkable benchmark for constraining the cosmological parameters, particularly     and    . The abundance of independent closed surfaces (voids) compared to the connected components (clusters) and independent loops (filaments), is more sensitive to the presence of     for   Mpc     irrespective of whether using the total matter density field ( ) or CDM+baryons field ( ). Reducing the degeneracy between     and     is achieved via Persistent Homology for the   field but not for the   field. The uncertainty of     at   confidence interval from the joint analysis of Persistent Homology vectorization for the   and   fields smoothed by   Mpc     at   reaches   eV and   eV, respectively. Noticing the use of the 3-dimensional underlying density field at  , the mentioned uncertainties can be treated as the theoretical lower limits