Anomalies in the topology of the temperature fluctuations in the cosmic
  microwave background: An analysis of the $\texttt{NPIPE}$ and
  $\texttt{FFP10}$ data releases
  We present a multi-scale topological analysis of the temperature fluctuation
maps from the NPIPE and FFP10 datasets, invoking relative homology to account
for the analysis in the presence of masks. For the topological components, we
detect a $2.96\sigma$ deviation between the observations and simulations at $N
= 128, FWHM = 80'$, for the FFP10 dataset. For the topological loops, we
observe a high deviation between the observation and simulations in the number
of loops at $FWHM = 320'$, at a low dimensionless threshold $\nu = -2.5$, for
the NPIPE dataset. Under a Gaussian assumption, this would amount to a
deviation of $\sim 4\sigma$ . However, the distribution in this bin is
manifestly non-Gaussian and does not obey Poisson statistics either. In the
absence of a true theoretical understanding, we simply note that the
significance is higher than what may be resolved by $600$ simulations. The
FFP10 dataset, indicates a $2.77\sigma$ deviation at this resolution and
threshold. The Euler characteristic reflects the deviations in the components
and loops. To assess the significance of combined levels for a given scale, we
employed the empirical and theoretical versions of the $\chi^2$ test as well as
the nonparametric Tukey depth test. Although all statistics exhibit a stable
distribution, we favor the empirical version of the $\chi^2$ test in the final
interpretation, as it indicates the most conservative differences. Even though
both datasets exhibit mild to significant discrepancies, they also exhibit
contrasting behaviors at various instances. Therefore, we do not find it
feasible to convincingly accept or reject the null hypothesis. Disregarding the
large-scale anomalies that persist at similar scales in WMAP and Planck,
observations of the cosmic microwave background are largely consistent with the
standard cosmological model within $2\sigma$.
