Past and Future of Cosmic Topology

The global topology of the universe, including questions about the shape of space, its volume and its connectedness, is a fundamental issue in cosmology which has been overlooked for many years, except by some pioneering authors (see references in LachièzeRey & Luminet, 1995, hereafter LaLu95). In the first part of the present article, I set out some unexplored historical material about the early development of cosmic topology. It stems out that the two “fathers” of the big bang concept, Friedmann and Lemâıtre, were also the first to realize the full importance of cosmic topology, whereas Einstein remained reluctant to the idea of a multi–connected space. In the second part I briefly comment new developments in the field since 1995, both from a theoretical and an observational point of view. They fully confirm that cosmic topology is, more than ever, a promising field of investigation. 1 Birth of cosmic topology One of the oldest cosmological questions is the physical extension of space : is it finite or infinite? (see e.g. Luminet, 1994 ; Luminet & Lachièze-Rey, 1994). In the history of cosmology, it is well known that the Newtonian physical space, mathematically identified with infinite Euclidean space R, gave rise to paradoxes such as darkness of night (see e.g. Harrison, 1987) and to problems of boundary conditions. Regarding for instance the Mach’s idea according to which local inertia would result from the contributions of masses at infinity, an obvious divergence difficulty arose, since a homogeneous Newtonian universe with non–zero density had an infinite mass. The aim of relativistic cosmology was to deduce from gravitational field equations some physical models of the universe as a whole. When Einstein (1917) assumed in his static cosmological solution that space was a positively–curved hypersphere, one of his strongest motivations was to provide a model for a finite space, although without a boundary. He regarded the closure of space as necessary to solve the problem of inertia (Einstein, 1934). The spherical model cleared up most of the paradoxes stemmed from Newtonian cosmology in such an elegant way that most cosmologists of the time adopted the new paradigm of a closed space, without examining other geometrical possibilities. Einstein was also convinced that the hypershere provided not only the metric of cosmic