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Homology stratifications and intersection homology

Colin Rourke and Brian Sanderson

Geometry & Topology Monographs 2 (1999) 455–472

DOI: 10.2140/gtm.1999.2.455

arXiv: math.GT/9911259


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.


permutation homology, intersection homology, homology stratification, homology general position

Mathematical Subject Classification

Primary: 55N33, 57Q25, 57Q65

Secondary: 18G35, 18G60, 54E20, 55N10, 57N80, 57P05


Received: 16 November 1998
Revised: 8 July 1999
Published: 21 November 1999

Colin Rourke
Mathematics Institute
University of Warwick
United Kingdom
Brian Sanderson
Mathematics Institute
University of Warwick
United Kingdom