01 · The idea
In everyday language
A manifold can be assembled by attaching elementary pieces called handles. Smale showed how, in sufficiently high dimensions, handles can be rearranged and cancelled under suitable hypotheses. His h-cobordism methods led to a proof that a smooth homotopy sphere of dimension at least five is homeomorphic to the ordinary sphere. The result links a global description by homotopy to an actual classification of the underlying topological space. High dimension supplies room for geometric manoeuvres that may fail in lower dimensions, making dimension an essential part of the argument rather than a cosmetic parameter.
Why it matters
Handle cancellation and h-cobordism became basic tools of high-dimensional topology, reorganising the classification of manifolds and inspiring further geometric methods.
