The Shape of Space between WMAP and PLANCK

Cosmic topology has entered a new era of experimental tests since accurate maps of the Cosmic Microwave Background radiation (CMB) are available. Various deviations from the concordance Λ-CDM flat infinite universe model hint at a possible non-trivial topology for the shape of space. A finite universe with the topology of the Poincaré dodecahedral spherical space fits remarkably well the data and is a good candidate for explaining both the local curvature of space and the large angle anomalies in the temperature power spectrum. Such a model of a ``small universe'', the volume of which would represent only about 80% the volume of the observable universe, offers an observational signature in the form of a predictable topological lens effect on one hand, and rises new issues on the physics of the early universe on the other hand.