  The standard cosmological model, which assumes statistical isotropy and
parity invariance, predicts the absence of correlations between even-parity and
odd-parity observables of the cosmic microwave background (CMB). Contrary to
these predictions, large-angle CMB temperature anomalies generically involve
correlations between even-$\ell$ and odd-$\ell$ angular power spectrum
$C_\ell$, while recent analyses of CMB polarization have revealed non-zero
equal-$\ell$ $EB$ correlations. These findings challenge the conventional
understanding, suggesting deviations from statistical isotropy, violations of
parity, or both. Cosmic topology, which involves changing only the boundary
conditions of space relative to standard cosmology, offers a compelling
framework to potentially account for such parity-violating observations.
Topology inherently breaks statistical isotropy, and can also break homogeneity
and parity, providing a natural paradigm for explaining observations of
parity-breaking observables without the need to add parity violation to the
underlying microphysics. Our investigation delves into the harmonic space
implications of topology for CMB correlations, using as an illustrative example
$EB$ correlations generated by tensor perturbations under both
parity-preserving and parity-violating scenarios. Consequently, these findings
not only challenge the foundational assumptions of the standard cosmological
model but also open new avenues for exploring the topological structure of the
Universe through CMB observations.
