Summary: The author studies a model which describes the birth of the universe using a global topological phase transition with a complex manifold where the time \(\tau\) is considered as a complex variable. Before the big bang, \(\tau\) is a purely imaginary variable so that the space can be considered as Euclidean. The phase transition from a pre-inflation to inflation is examined by studying the dynamical rotation of the time on the complex plane. Back-reaction effects are exactly calculated using relativistic quantum geometry.