  The early history of the universe might be described by a topological phase
followed by a standard second phase of Einstein gravity. To study this scenario
in its full generality, we consider a four-manifold of Euclidean signature in
the topological phase, which shares a common boundary with a corresponding
manifold of Lorentzian signature in the Einstein phase. We find that the
boundary should have vanishing extrinsic curvature, whereas the manifold in the
topological phase should have zero Euler number. In addition, we show that the
second phase must be characterized by an initial vanishing Weyl tensor and that
the standard cosmological flatness problem is not automatically solved unless a
conformal invariant boundary term is added. We also characterize the scalar
perturbations in the standard Einstein phase. We show that they must contain an
initial non-vanishing shear component inherited from the topological phase and
we estimate the non-Gaussian parameters. Finally, we argue that the topological
early universe cosmology shares common features of previous ideas, such as the
so-called Weyl curvature hypothesis, the universe's creation out of nothing and
the no-boundary proposal.
