  In this paper we discuss a space-time having the topology of S^{3}xR but with
different smoothness structure. This space-time is not a global hyperbolic
space-time. Especially we obtain a time line with a topology change of the
space from the 3-sphere to a homology 3-sphere and back but without a
topology-change of the space-time. Among the infinite possible smoothness
structures of this space-time, we choose a homology 3-sphere with hyperbolic
geometry admitting a homogenous metric. Then the topology change can be
described by a time-dependent curvature parameter k changing from k=+1 to k=-1
and back. The solution of the Friedman equation for dust matter (p=0) after
inserting this function shows an exponential growing which is typical for
inflation. In contrast to other inflation models, this process stops after a
finite time.
