arxiv: 2604.14913 · v1 · submitted 2026-04-16 · 🧮 math.AT · math.KT

Recognition: unknown

Equivariant L-Classes of Atiyah-Singer-Zagier Type for Singular Spaces

Markus Banagl

Pith reviewed 2026-05-10 10:15 UTC · model grok-4.3

classification 🧮 math.AT math.KT

keywords Witt pseudomanifoldsequivariant L-classesGoresky-MacPherson L-classesfinite group actionssingular spacesintersection homologycharacteristic classesorbit spaces

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The pith

Finite group actions on Witt pseudomanifolds yield orbit spaces that are also Witt pseudomanifolds, with equivariant L-classes averaging to recover the Goresky-MacPherson L-class of the orbit space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that a finite group action on a Witt pseudomanifold produces an orbit space that remains a Witt pseudomanifold. This holds for spaces including all complex pure-dimensional algebraic varieties. As a result, the orbit space carries Goresky-MacPherson L-classes. The authors introduce equivariant L-classes of Atiyah-Singer-Zagier type on the original G-space. An averaging formula then shows these equivariant classes compute the L-class on the orbit space. The construction uses intersection homology transfer properties together with a G-signature theorem.

Core claim

If a finite group G acts on a Witt pseudomanifold, the orbit space is again a Witt pseudomanifold. In the compact oriented case this guarantees Goresky-MacPherson L-classes on the orbit space. Equivariant L-classes of Atiyah-Singer-Zagier type are constructed on the G-pseudomanifold using intersection-homological transfer maps and the G-signature theorem; an averaging formula recovers the Goresky-MacPherson L-class of the orbit space from these equivariant classes.

What carries the argument

Atiyah-Singer-Zagier type equivariant L-classes on G-pseudomanifolds, built from intersection homology transfers and the G-signature theorem.

If this is right

Orbit spaces of finite group actions on Witt pseudomanifolds are themselves Witt pseudomanifolds.
Goresky-MacPherson L-classes are defined on these orbit spaces.
The averaging formula gives a direct computational link from the equivariant classes to the orbit-space L-class.
The result applies to all compact oriented complex pure-dimensional algebraic varieties with finite group actions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method supplies a route to other equivariant characteristic classes on singular spaces once the Witt condition is preserved.
One could verify the averaging formula on explicit orbifold examples with isolated singularities.
The dependence on prior G-signature results suggests the construction integrates with K-theoretic approaches to index theory on pseudomanifolds.

Load-bearing premise

The starting space must be a Witt pseudomanifold and the finite group action must preserve the Witt condition so that the orbit space remains Witt.

What would settle it

A concrete Witt pseudomanifold equipped with a finite group action whose orbit space fails the Witt condition, or an explicit computation where the averaging formula does not recover the Goresky-MacPherson L-class of the orbit space.

read the original abstract

If a finite group $G$ acts on a rational homology manifold, then the orbit space is well-known to be a rational homology manifold again. We consider here actions on spaces that may be much more singular. If the $G$-space is a Witt pseudomanifold, which includes all arbitrarily singular complex pure-dimensional algebraic varieties, then we prove that the orbit space is again a Witt pseudomanifold. In the compact oriented situation, this implies that the orbit space possesses characteristic L-classes, as defined by Goresky and MacPherson. We then construct Atiyah-Singer-Zagier type equivariant L-classes for such $G$-pseudomanifolds which serve, as we show by establishing an averaging formula, as a tool to compute the Goresky-MacPherson L-class of the orbit space. The construction of the equivariant class builds on intersection homological transfer properties and on recent joint K-theoretic work with Eric Leichtnam and Paolo Piazza, which established a G-signature theorem on Witt pseudomanifolds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows finite group actions preserve the Witt pseudomanifold property and gives an averaging formula from equivariant Atiyah-Singer-Zagier L-classes down to the Goresky-MacPherson L-class on the orbit space.

read the letter

The core result is that a finite group action on a Witt pseudomanifold keeps the orbit space a Witt pseudomanifold, so Goresky-MacPherson L-classes exist on the quotient. They construct equivariant L-classes of Atiyah-Singer-Zagier type for the G-space and prove these average to the L-class of the orbit space. This is the concrete new piece: the equivariant construction plus the averaging formula for these singular spaces under group actions. It applies directly to complex algebraic varieties since those are Witt pseudomanifolds. The approach uses intersection homology transfer maps and rests on the authors' earlier G-signature theorem with Leichtnam and Piazza. That prior result is cited as external, so the dependence is normal rather than circular. The high-level setup matches standard techniques for equivariant stratified spaces, and the stress test finds no internal gaps in the stated architecture. The Witt preservation claim is the part that needs the most careful checking in the full proofs, since the abstract only states the outcome. If the transfer properties and signature theorem apply cleanly, the averaging formula should give a usable computational tool. This is for people already working in intersection homology, equivariant topology, or K-theory of singular spaces with symmetries. A reader who needs explicit ways to compute L-classes on quotients of varieties or pseudomanifolds will find the formulas useful. It deserves a serious referee because the claims are specific, the methods are grounded, and the potential applications are clear even if the proofs require verification.

Referee Report

2 major / 2 minor

Summary. The paper proves that if a finite group G acts on a Witt pseudomanifold (including all complex pure-dimensional algebraic varieties), then the orbit space is again a Witt pseudomanifold, so that Goresky-MacPherson L-classes exist on the orbit space in the compact oriented case. It constructs Atiyah-Singer-Zagier type equivariant L-classes for such G-pseudomanifolds via intersection-homological transfer maps, relying on a prior K-theoretic G-signature theorem, and establishes an averaging formula showing that these equivariant classes compute the Goresky-MacPherson L-class of the orbit space.

Significance. If the results hold, the work extends equivariant characteristic class theory from manifolds to highly singular stratified spaces with finite group actions, providing a concrete computational tool via the averaging formula. The preservation of the Witt property is foundational for applying Goresky-MacPherson theory to quotients, and the construction integrates intersection homology with K-theoretic methods, strengthening links to existing literature on singular spaces and equivariant signatures.

major comments (2)

[Abstract and §2 (or wherever the preservation is proved)] The preservation of the Witt pseudomanifold property under finite group actions is load-bearing for the existence of Goresky-MacPherson L-classes on the orbit space (as stated in the abstract and used throughout). The manuscript should supply explicit verification steps or a dedicated subsection detailing how the group action preserves the Witt condition via intersection homology, rather than asserting it at a high level.
[Section on the averaging formula (likely §4 or §5)] The averaging formula (central to the utility of the equivariant L-classes) depends on the G-signature theorem from the authors' prior joint work with Leichtnam and Piazza. A precise citation to the specific theorem or equation from that work, together with a brief recall of the statement as applied here, is needed to make the derivation self-contained and to clarify the logical dependence.

minor comments (2)

Clarify the notation for the equivariant L-classes (e.g., how they are denoted versus the ordinary Goresky-MacPherson classes) and ensure consistent use of transfer maps throughout.
Add a short remark on the scope: whether the results extend beyond compact oriented cases or require additional hypotheses on the group action.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation for minor revision. The comments are helpful for improving clarity, and we address each major point below.

read point-by-point responses

Referee: [Abstract and §2 (or wherever the preservation is proved)] The preservation of the Witt pseudomanifold property under finite group actions is load-bearing for the existence of Goresky-MacPherson L-classes on the orbit space (as stated in the abstract and used throughout). The manuscript should supply explicit verification steps or a dedicated subsection detailing how the group action preserves the Witt condition via intersection homology, rather than asserting it at a high level.

Authors: We agree that greater explicitness will strengthen the exposition. The proof that finite group actions preserve the Witt property is given in Section 2, relying on the definition of Witt pseudomanifolds via vanishing of middle-dimensional intersection homology and the fact that the group action induces isomorphisms on the relevant local intersection homology groups. To address the comment, we will insert a dedicated subsection in Section 2 that spells out the verification steps in detail, including the precise local conditions and how they are preserved. revision: yes
Referee: [Section on the averaging formula (likely §4 or §5)] The averaging formula (central to the utility of the equivariant L-classes) depends on the G-signature theorem from the authors' prior joint work with Leichtnam and Piazza. A precise citation to the specific theorem or equation from that work, together with a brief recall of the statement as applied here, is needed to make the derivation self-contained and to clarify the logical dependence.

Authors: We thank the referee for this observation. We will add an explicit citation to the relevant theorem (the G-signature theorem for Witt pseudomanifolds) from our prior joint work with Leichtnam and Piazza, together with a short paragraph recalling its statement in the form needed for the averaging argument. This will make the logical dependence transparent and the derivation self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation establishes that finite group actions preserve the Witt pseudomanifold property (a new result for singular spaces including algebraic varieties) and constructs equivariant L-classes via intersection-homological transfers plus an averaging formula that computes the Goresky-MacPherson L-class on the orbit space. The reference to the G-signature theorem from prior joint work with Leichtnam and Piazza supplies an external theorem as a building block rather than creating an internal loop; the current paper's proofs and constructions do not reduce by definition or construction to that prior result or to any self-referential input. No self-definitional steps, fitted predictions, or load-bearing self-citations that collapse the argument are present.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard definition of Witt pseudomanifolds from intersection homology theory and on the G-signature theorem established in the authors' prior joint work; no free parameters or new invented entities are introduced.

axioms (2)

domain assumption Finite group actions on Witt pseudomanifolds yield orbit spaces that are again Witt pseudomanifolds
This is the key preservation statement asserted in the abstract as the foundation for defining L-classes on the orbit space.
domain assumption The G-signature theorem holds for Witt pseudomanifolds as proved in prior joint work
Invoked as the K-theoretic foundation for constructing the equivariant L-classes and the averaging formula.

pith-pipeline@v0.9.0 · 5477 in / 1542 out tokens · 50054 ms · 2026-05-10T10:15:04.273805+00:00 · methodology

discussion (0)

Reference graph

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