  In this paper we analyze the long-time behaviour of 3 dimensional Ricci flow
with surgery. We prove that under the topological condition that the initial
manifold only has non-aspherical or hyperbolic components in its geometric
decomposition, there are only finitely many surgeries and the curvature is
bounded by $C t^{-1}$ for large $t$. This answers an open question in
Perelman's work, which was made more precise by Lott and Tian, for this class
of initial topologies.
  More general classes will be discussed in subsequent papers using similar
methods.
