Cosmic Topology
  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. 
Jaspreet Sandhu
astro-ph.CO