  Exploiting the Persistent Homology technique and an associated complementary
representation which enables us to construct a synergistic pipeline for
different topological features quantified by Betti curves in reducing the
degeneracy between cosmological parameters, we investigate the footprint of
summed massive neutrinos ($M_{\nu}$) in different density fields simulated by
the publicly available Quijote suite. Evolution of topological features in the
context of super-level filtration on three-dimensional density fields, reveals
remarkable indicators for constraining the $M_{\nu}$ and $\sigma_8$. The
abundance of 2-holes is more sensitive to the presence of $M_{\nu}$, also the
persistence of topological features plays a crucial role in cosmological
inference and reducing the degeneracy associated with $M_{\nu}$ simulation
rather than their birth thresholds when either the total matter density ($m$)
field or those part including only cold dark matter+baryons ($cb$) is utilized.
Incorporating the Betti-1 and Betti-2 for $cb$ part of $M^+_{\nu}$ simulation
marginalized over the thresholds implies $5\%$ variation compared to the
massless neutrinos simulation. The constraint on $M_{\nu}$ from $\beta_k$ and
its joint analysis with birth threshold and persistency of topological features
for total mass density field smoothed by $R=5$ Mpc h$^{-1}$ at zero redshift
reach to $0.0172$ eV and $0.0152$ eV, at $1\sigma$ confidence interval,
respectively.
