Intersection homotopy, refinements and coarsenings
Pith reviewed 2026-05-19 01:03 UTC · model grok-4.3
The pith
Intersection homotopy groups of CS sets stay the same after coarsening when the regular part is unchanged and the perversity grows.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For two CS-set structures on the same topological space in which the strata of one are unions of the strata of the other, the intersection homotopy groups computed from a general perversity on the finer structure coincide with those computed from its pushforward on the coarser structure, provided the regular parts coincide and the perversity satisfies the growing property.
What carries the argument
Pushforward of a general perversity (defined on the poset of strata) under coarsening of CS sets, which transfers the intersection homotopy groups defined on Gajer's simplicial sets.
If this is right
- Intersection homotopy groups become independent of the choice of stratification inside a fixed regular part when the growing condition holds.
- The groups can be compared across different but compatible stratifications of Thom-Mather spaces when singular strata become regular.
- The invariance supplies a tool for reducing computations to a convenient coarsening while keeping the same homotopy data.
Where Pith is reading between the lines
- Computations of these groups could be simplified by always moving to the coarsest possible stratification that preserves the regular part.
- The result may extend the range of spaces on which intersection homotopy groups can be defined and compared without changing the underlying topology.
Load-bearing premise
The regular part of the space must be identical for both CS-set structures, and the general perversity must satisfy the growing property analogous to the original Goresky-MacPherson condition.
What would settle it
An explicit pair of coarsened CS-set structures with identical regular parts, a general perversity lacking the growing property, and a direct computation showing that the intersection homotopy groups differ.
read the original abstract
In previous works, we studied intersection homotopy groups associated to a Goresky and MacPherson perversity and a filtered space. They are defined as the homotopy groups of simplicial sets introduced by P. Gajer. We particularized to locally conical spaces of Siebenmann (called CS sets) and established a topological invariance for them when the regular part remains unchanged. Here, we consider coarsenings, made of two structures of CS sets on the same topological space, the strata of one being a union of strata of the other. We endow them with a general perversity and its pushforward, where the adjective ``general'' means that the perversities are defined on the poset of the strata and not only according to their codimension. If the perversity verifies a growing property analogous to that of the original perversities of Goresky and MacPherson, we also find an invariance theorem for the intersection homotopy groups of a coarsening, under the above restriction on the regular parts. An invariance is shown too in some cases where singular strata become regular in the coarsening, for Thom-Mather spaces.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper extends intersection homotopy groups, previously defined via Goresky-MacPherson perversities and Gajer's simplicial sets on filtered spaces and CS sets, to general perversities defined on the poset of strata rather than solely by codimension. It introduces pushforwards of such perversities under coarsenings (where one CS set structure refines the other on the same topological space) and proves invariance of the resulting intersection homotopy groups when the perversity satisfies a growing property analogous to the classical case and the regular part is preserved. Additional invariance results are established for Thom-Mather spaces in cases where singular strata become regular under the coarsening.
Significance. If the invariance theorems hold, the work provides a useful generalization that allows perversities to depend on the specific stratification rather than codimension alone, facilitating comparisons across different CS set structures on the same space. The explicit use of the growing property and preservation of the regular part yields precise, checkable conditions for invariance, extending prior topological invariance results. The treatment of Thom-Mather spaces adds applicability to spaces with conical singularities. These contributions strengthen the toolkit for computing homotopy invariants of singular spaces.
major comments (1)
- [Invariance theorem for coarsenings] The central invariance theorem for coarsenings (stated in the abstract and presumably proved in the section following the definition of pushforward) relies on the growing property of the general perversity and the unchanged regular part. The manuscript should explicitly verify that the simplicial sets of Gajer yield isomorphic homotopy groups under the pushforward map when these hold; without a detailed comparison of the face and degeneracy maps in the relevant simplicial sets, the reduction to the prior invariance result remains somewhat opaque.
minor comments (2)
- [Preliminaries] Notation for the poset of strata and the pushforward operation could be clarified with a small diagram or explicit formula in the preliminaries section to aid readers unfamiliar with general perversities.
- [Thom-Mather spaces section] The abstract mentions 'some cases' for Thom-Mather spaces; a precise characterization of those cases (e.g., local conical conditions) would strengthen the statement.
Simulated Author's Rebuttal
We thank the referee for the careful reading, positive assessment of the significance, and recommendation for minor revision. We address the major comment below.
read point-by-point responses
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Referee: [Invariance theorem for coarsenings] The central invariance theorem for coarsenings (stated in the abstract and presumably proved in the section following the definition of pushforward) relies on the growing property of the general perversity and the unchanged regular part. The manuscript should explicitly verify that the simplicial sets of Gajer yield isomorphic homotopy groups under the pushforward map when these hold; without a detailed comparison of the face and degeneracy maps in the relevant simplicial sets, the reduction to the prior invariance result remains somewhat opaque.
Authors: We agree that greater explicitness would strengthen the exposition. The argument reduces the invariance to our prior result on CS sets by constructing a simplicial map between the Gajer simplicial sets associated to the refined and coarsened structures. The growing condition on the perversity ensures that a simplex is admissible for the pushforward perversity precisely when its image under the coarsening satisfies the original conditions, while preservation of the regular part allows direct application of the earlier homotopy equivalence. To address the opacity, we will insert a short lemma (or expanded remark) in the revised manuscript that explicitly checks compatibility of the face and degeneracy operators with the pushforward on strata, confirming the induced map on simplicial sets is well-defined and yields isomorphic homotopy groups under the stated hypotheses. revision: yes
Circularity Check
Minor self-citation of prior definitions; central invariance claims remain independent
full rationale
The derivation begins from prior definitions of intersection homotopy groups via Gajer's simplicial sets on CS sets (explicitly referenced in the abstract as 'in previous works'). The new invariance theorem for coarsenings is conditioned on two explicitly stated requirements: the growing property of the general perversity (analogous to Goresky-MacPherson) and preservation of the regular part. These conditions are applied directly to show that the simplicial sets yield the same homotopy groups before and after pushforward. The separate case for Thom-Mather spaces where singular strata become regular follows from local conical controls. No equation or claim reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the central results add independent content under the stated hypotheses. This is the expected pattern for an extension paper and warrants only a minimal circularity score.
Axiom & Free-Parameter Ledger
axioms (3)
- standard math Homotopy groups of simplicial sets associated to filtered spaces with perversities are well-defined
- domain assumption CS sets are locally conical filtered spaces with topological invariance under fixed regular part
- ad hoc to paper General perversities on poset of strata admit a pushforward and satisfy a growing property
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
If the perversity verifies a growing property analogous to that of the original perversities of Goresky and MacPherson, we also find an invariance theorem for the intersection homotopy groups of a coarsening, under the above restriction on the regular parts.
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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