Seminar Talk Milnor’s Counterexample to the Hauptvermutung

This section is based on the Hauptvermutung book [RCS96] edited by Andrew Ranicki. The word Hauptvermutung is a short form of the german term Hauptvermutung der kombinatorischen Topologie which means the main conjecture of combinatorial topology. It was first stated in 1908 by Tietze [Tie08] and by Steinitz [Ste08]. Their formulation was slightly different from the modern one, which is due to Rourke and Sanderson [RS70]. Before the appearing of Milnor’s article, the Hauptvermutung had been proven for manifolds of dimension at most 3 and for polyhedra of dimension at most 2. Milnor then gave counterexamples for all dimensions ≥ 3. His construction was later generalized by Stallings [Sta65]. However, these counterexamples were not manifolds, so the question arose, whether the Hauptvermutung was true for manifolds. This was called the manifold Hauptvermutung (and the original one is often called polyhedral Hauptvermutung to distinguish the two). But also the manifold Hauptvermutung was proven not to hold, which was done by Kirby and Siebenmann in 1969 [KS77].