Topology
Present "catalogs" of astrophysical objects (quasars, gamma-ray bursts and the CMB temperature fluctuations) have angular coordinates accurately measured which can be used to study the topological properties of the Universe. An interesting question which arises during these studies is if one can distinguish a possible cosmological pattern or topological signature imprinted in these catalogs from pure statistical fluctuations.
The study of the large-scale (i.e. global) homogeneity and isotropy of the Universe encompasses its topology as well, since topological properties are global properties of 3-spaces. Multiply-connectedness is a topological property that tesselates a simply-connected space (like the Euclidean space R3) producing multiple images of a given object. This property generates angular and distance correlations (i.e. anisotropies) in the objects distribution.
These can be studied mainly through the search of repeated images in the CMB anisotropy maps (e.g. the so-called "circles in the sky" method [23]), asymmetries in the amplitude of CMB large scale fluctuations [7], and the PASH method [4], which consists of temperature histograms from different regions of the sky.