  In this paper we will discuss a model which describes the cause of inflation
by a topological transition. The guiding principle is the choice of an exotic
smoothness structure for the space-time. Here we consider a space-time with
topology $S^{3}\times\mathbb{R}$. In case of an exotic $S^{3}\times\mathbb{R}$,
there is a change in the spatial topology from a 3-sphere to a homology
3-sphere which can carry a hyperbolic structure. From the physical point of
view, we will discuss the path integral for the Einstein-Hilbert action with
respect to a decomposition of the space-time. The inclusion of the boundary
terms produces fermionic contributions to the partition function. The
expectation value of an area (with respect to some surface) shows an
exponential increase, i.e. we obtain inflationary behavior. We will calculate
the amount of this increase to be a topological invariant. Then we will
describe this transition by an effective model, the Starobinski or $R^{2}$
model which is consistent with the current measurement of the Planck satellite.
The spectral index and other observables are also calculated. Finally we obtain
a realistic cosmological constant.
