  In this paper, we will consider a cosmological model with two topological
transitions of the space. The smooth 4-dimensional spacetime of the model
admits topological transitions of its 3-dimensional slices. The whole approach
is inspired by a class of exotic smoothness structure on
$S^{3}\times\mathbb{R}$. In particular, this class of smoothness structures
induces two topological transitions. Then, we are able to calculate the energy
scales as associated to these topological transitions. For the first transition
we will get the value of the GUT scale and the energy of the second transition
is at the electroweak scale. The topology of the exotic $S^{3}\times\mathbb{R}$
determines both, the energy of the scales by certain topological invariants,
and the existence of the right-handed sterile neutrino. It is the input for the
seesaw mechanism. Secondly, based on this model, we are able to calculate the
neutrino masses which are in a very good agreement with experiments. Finally,
we will speculate, again based on topology, why there are three generations of
neutrinos and an asymmetry between neutrinos and anti-neutrinos.
