  In this paper we discuss a spacetime having the topology of S^3 x R but with
a different smoothness structure leading to a geometric model for inflation,
called geometric inflation. In particular this spacetime is not globally
hyperbolic and we obtain a time line with a spatial topology change from the
3-sphere to a homology 3-sphere and back. The topology of the spacetime remains
invariant. Among the infinite possible smoothness structures of this spacetime,
we choose a homology 3-sphere constructed from the knot 8_{10} with hyperbolic
geometry, i.e. admitting a homogenous metric of negative scalar curvature. We
discuss the accelerated expansion for FLRW cosmology caused by the topology
change. In contrast to other inflation models, this process stops after a
finite time. Alternatively, the topology change can be also described by a
SU(2)-valued scalar field. Then we calculate the expansion rate (having more
than 60 e-folds) and the energy time scale. The coupling to matter is also
interpreted geometrically and the reheating process (as well the supercooled
expansion during inflation) is naturally obtained. The model depends only on a
single parameter, a topological invariant of the homology 3-sphere, and assumes
a Planck size universe of S^3-topology. The dependence of the model on the
initial state and the a geometric interpretation of quantum fluctuations are
also discussed.
