Regarding your first question, in 1953 Moise proved the (manifold) Hauptvermutung for 3-manifolds (Ann. of Math. 58, pp. 458-480). One way to state his result is that every homeomorphism (diffeomorphism) between compact 3-manifolds is homotopic to a PL homeomorphim (simple homotopy equivalence)homeomorphism.
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Regarding your first question, in 1953 Moise proved the (manifold) Hauptvermutung for 3-manifolds (Ann. of Math. 58, pp. 458-480). One way to state his result is that every homeomorphism (diffeomorphism) between compact 3-manifolds is homotopic to a PL homeomorphim (simple homotopy equivalence). |
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