Abstract
This is an expository article with complete proofs intended for a general nonspecialist 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.
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