Alexandrov spaces are CS sets

Abstract

We prove that the extremal stratification of an Alexandrov space introduced by Perelman-Petrunin is a CS stratification in the sense of Siebenmann. We also show that every space of directions of an Alexandrov space without proper extremal subsets is homeomorphic to a sphere. In the polyhedral case, the same holds for every iterated space of directions.

arXiv commentrevised incorrect terminology on PL structure in Theorem 1.4 and Lemma 5.1 and added Remarks 1.5 and 1.7