  This is an expository article with complete proofs intended for a general
non-specialist audience. The results are two-fold. First, we discuss a
geometric invariant, that we call the width, of a manifold and show how it can
be realized as the sum of areas of minimal 2-spheres. For instance, when $M$ is
a homotopy 3-sphere, the width is loosely speaking the area of the smallest
2-sphere needed to ``pull over'' $M$. Second, we use this to conclude that
Hamilton's Ricci flow becomes extinct in finite time on any homotopy 3-sphere.
We have chosen to write this since the results and ideas given here are quite
useful and seem to be of interest to a wide audience.
