A reasonable notion of dimension for singular intersection homology

TL;DR

This paper demonstrates that polyhedral and skeleta-based notions of dimension yield equivalent intersection homology theories on Siebenmann's CS sets, establishing polyhedral dimension as a reasonable choice.

Contribution

It proves the isomorphism of intersection homologies derived from two different dimension notions, using a Mayer-Vietoris argument with a novel subdivision technique.

Findings

Polyhedral and skeleta-based dimensions produce isomorphic intersection homologies.

A new subdivision method with pseudo-barycentric points facilitates the proof.

Polyhedral dimension is validated as a reasonable dimension for intersection homology.

Abstract

M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for subspaces $S$ of a Euclidean simplex, which is usually taken as the smallest dimension of the skeleta containing~$S$. Later, P. Gajer employed another dimension based on the dimension of polyhedra containing $S$. This last one allows traces of pullbacks of singular strata in the interior of the domain of a singular simplex. In this work, we prove that the two corresponding intersection homologies are isomorphic for Siebenmann's CS sets. In terms of King's paper, this means that polyhedral dimension is a ``reasonable'' dimension. The proof uses a Mayer-Vietoris argument which needs an adaptated subdivision. With the polyhedral dimension, that is a subtle issue. General position arguments are not sufficient and we introduce strong general position. With it, a stability is added to the generic character and we can do an inductive cutting of each singular simplex. This decomposition is realised with pseudo-barycentric subdivisions where the new vertices are not barycentres but close points of them.