A homology stratification is a filtered space with local homology groups constant on
strata. Despite being used by Goresky and MacPherson [Intersection homology
theory II, Inventiones Mathematicae, 71 (1983) 77–129] in their proof of topological
invariance of intersection homology, homology stratifications do not appear to have
been studied in any detail and their properties remain obscure. Here we use them to
present a simplified version of the Goresky–MacPherson proof valid for PL spaces,
and we ask a number of questions. The proof uses a new technique, homology general
position, which sheds light on the (open) problem of defining generalised intersection
homology.
Rob Kirby has been a great source of
encouragement. His help in founding the new electronic
journal Geometry & Topology has been invaluable. It is a
great pleasure to dedicate this paper to him.
Keywords
permutation homology, intersection
homology, homology stratification, homology general
position
