On the triangulation of stratified sets and singular varieties

Abstract

We show that every compact stratified set in the sense of Thom can be triangulated as a simplicial complex. The proof uses that author’s description of a stratified set as the geometric realisation of a certain type of diagram of smooth fibre bundles and smooth imbeddings, and the triangulability of smooth fibre bundles. As a consequence, we obtain proofs of the classical triangulation theorems for analytic and subanalytic sets, and a correct proof of Yang’s theorem that the orbit space of a smooth compact transformation group is triangulable.