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A reasonable notion of dimension for singular intersection homology

David Chataur, Martintxo Saralegi-Aranguren, Daniel Tanré

TL;DR

This work compares two definitions of intersection homology arising from different dimension notions—King’s chain-based approach and Gajer’s polyhedral-dimension criterion—showing they yield the same groups on Siebenmann CS sets. The authors develop a robust framework built on strong general position and pseudo-barycentric subdivisions to preserve admissibility under subdivision, enabling Mayer–Vietoris arguments within the perverse setting. Central contributions include a precise cone formula for open cones, the existence of subdivision techniques that respect perverse allowability, and a proof that polyhedral-dimension IH is isomorphic to King’s IH via a transformation theorem. The results validate the polyhedral-dimension approach as a reasonable and equivalent foundation for intersection homology in this generality, with implications for both theory and applications to singular spaces.

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.

A reasonable notion of dimension for singular intersection homology

TL;DR

This work compares two definitions of intersection homology arising from different dimension notions—King’s chain-based approach and Gajer’s polyhedral-dimension criterion—showing they yield the same groups on Siebenmann CS sets. The authors develop a robust framework built on strong general position and pseudo-barycentric subdivisions to preserve admissibility under subdivision, enabling Mayer–Vietoris arguments within the perverse setting. Central contributions include a precise cone formula for open cones, the existence of subdivision techniques that respect perverse allowability, and a proof that polyhedral-dimension IH is isomorphic to King’s IH via a transformation theorem. The results validate the polyhedral-dimension approach as a reasonable and equivalent foundation for intersection homology in this generality, with implications for both theory and applications to singular spaces.

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 of a Euclidean simplex, which is usually taken as the smallest dimension of the skeleta containing~. Later, P. Gajer employed another dimension based on the dimension of polyhedra containing . 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.
Paper Structure (17 sections, 19 theorems, 72 equations, 1 table)

This paper contains 17 sections, 19 theorems, 72 equations, 1 table.

Table of Contents

  1. Outline of the paper
  2. Notation.
  3. Some reminders
  4. Intersection homology
  5. Dimension of polyhedra
  6. Chain maps induced by stratified maps
  7. The case of a cone
  8. Strong general position
  9. General position
  10. Strong general position
  11. Pseudo-barycentric subdivisions
  12. Existence of pseudo-barycentric subdivisions
  13. Homotopy of pseudo-barycentric subdivision
  14. Mayer-Vietoris sequence
  15. Intersection homology is intersection homology
  16. ...and 2 more sections

Key Result

Proposition 2.5

Let $f\colon (X,\overline{p})\to (Y,\overline{q})$ be a stratified map between two perverse spaces with $f^*D\overline q \leq D\overline p$. The association $\sigma\mapsto f\circ \sigma$ sends a $\overline{p}$-allowable simplex on a $\overline{q}$-allowable simplex and defines a chain map $f_{*}\col

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 45 more