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I think it is reasonable to view the h-cobordism theorem (and its relative, the s-cobordism theorem) as a workhorse of differential topology. The statement isn't actually that hard - it simply gives a natural condition under which a cobordism between two high dimensional manifolds is homotopically trivial - but the proof is quite difficult (it implies the high dimensional Poincare conjecture and contributed greatly to Smale's Fields medal). And the theorem is at the very heart of the surgery exact sequence, one of the most important and powerful tools in high dimensional topology.