Differential Topology 46 Years Later

The first big breakthrough, by [Kirby-Siebenmann, 1969, 1969a, 1977], was an obstruction theory for the problem of triangulating a given topological manifold as a PL (= Piecewise-Linear) manifold. If BTop and BPL are the stable classifying spaces (as described in the lectures), they showed that the relative homotopy group πj(BTop, BPL) is cyclic of order two for j = 4, and zero otherwise. Given an n-dimensional topological manifold M, it follows that there is an obstruction o ∈ H(M; Z/2) to triangulating M as a PL-manifold. In dimensions n ≥ 5 this is the only obstruction. Given such a triangulation, there is a similar obstruction in H(M; Z/2) to its uniqueness up to a PL-isomorphism which is topologically isotopic to the identity. In particular, they proved the following.