  This review aims to cover the central aspects of current research in cosmic
topology from a topological and observational perspective. Beginning with an
overview of the basic concepts of cosmology, it is observed that though a
determinant of local curvature, Einstein's equations of relativity do not
constrain the global properties of space-time. The topological requirements of
a universal space time manifold are discussed, including requirements of
space-time orientability and causality. The basic topological concepts used in
classification of spaces, i.e. the concept of the Fundamental Domain and
Universal covering spaces are discussed briefly. The manifold properties and
symmetry groups for three dimensional manifolds of constant curvature for
negative, positive and zero curvature manifolds are laid out.
Multi-connectedness is explored as a possible explanation for the detected
anomalies in the quadrupole and octopole regions of the power spectrum,
pointing at a possible compactness along one or more directions in space. The
statistical significance of the evidence, however, is also scrutinized and I
discuss briefly the bayesian and frequentist interpretation of the posterior
probabilities of observing the anomalies in a Lambda CDM universe. Some of the
major topologies that have been proposed and investigated as possible
candidates of a universal manifold are the Poincare Dodecahedron and Bianchi
Universes, which are studied in detail. Lastly, the methods that have been
proposed for detecting a multi connected signature are discussed. These include
ingenious observational methods like the circles in the sky method, cosmic
crystallography and Bayesian Analysis which provides the additional advantage
of being free from measurement errors and uses the posterior likelihoods of
models. As of the recent Planck mission, no pressing evidence of a multi
connected topology has been detected.
