Topology of the universe and the cosmic microwave background radiation

Recent astronomical observations of recession of further and further galaxies and clusters of galaxies, estimates of masses of clusters of galaxies, studies of temperature anisotropies of cosmic microwave back-round radiation, and type Ia supernovae suggest that the universe is flat (Euclidean). However the geometry of space does not determine its shape, for example a 3 dimensional Euclidean space admits 18 different topologies. Here we present the basic observational data indicating that the universe is flat or almost flat. Possible compact (finite volume) topologies of 3D Euclidean space are presented and finally observations that could reveal the shape of the universe are discussed.