Cheeger–Goresky–MacPherson Conjecture

• The Cheeger–Goresky–MacPherson conjecture is a framework relating analytic L² de Rham cohomology with middle perversity intersection cohomology on stratified spaces.
• It refines classical cohomology comparisons by introducing mezzoperversities and sheaf-theoretic models to address complications on non-Witt spaces.
• The theory underpins signature, duality, and self-dual structures in Cheeger spaces, unifying analytic and topological approaches.

Searching arXiv for the core papers on the Cheeger–Goresky–MacPherson conjecture and its refinements. The Cheeger–Goresky–MacPherson conjecture concerns the relation between analytic $L^2$ de Rham cohomology on the regular part of a stratified pseudomanifold and the topological intersection cohomology introduced by Goresky and MacPherson. In its classical form, for a suitably stratified pseudomanifold $X$ equipped with an adapted metric $g$ on its regular part, the conjecture asserts that the reduced $L^2$ de Rham cohomology coincides with middle perversity intersection cohomology $IH^{\bar m,*}(X)$ (Albin et al., 2013). In the Witt case this is the established Cheeger–Goresky–MacPherson theorem, while on non-Witt spaces the conjecture requires refinement by additional boundary data, now formulated in terms of mezzoperversities (Albin et al., 2013). Subsequent work identifies the analytic $L^2$ theory with refined sheaf-theoretic models, develops the signature and Novikov packages on Cheeger spaces, and, in several complex-analytic settings, proves corresponding $L^2$-to-intersection-cohomology identifications (Albin et al., 2013, Albin, 2016, Shentu et al., 2021).

1. Classical formulation and the Witt-space theorem

Classically, the conjecture is stated for a suitably stratified pseudomanifold $X$ with an adapted metric $g$ on its regular part. The expected identification is between reduced $L^2$ de Rham cohomology and middle perversity intersection cohomology: $X$0 Here the lower middle perversity is

$X$1

and the upper middle perversity is

$X$2

with $X$3 and $X$4 complementary (Albin et al., 2013).

The established Witt case is the Cheeger–Goresky–MacPherson theorem. If $X$5 is a Witt space, then Cheeger’s $X$6 de Rham theory matches Goresky–MacPherson middle perversity intersection cohomology: $X$7 A space is Witt if for every stratum of odd codimension the middle-degree intersection homology of its link vanishes; equivalently, the problematic middle-degree $X$8 obstructions vanish, so no extra boundary conditions are needed and $X$9 has a unique closed extension (Albin et al., 2013).

The Witt/non-Witt distinction is the decisive structural divide. On Witt spaces, there is a unique middle perversity Deligne sheaf, Poincaré duality holds, and Cheeger’s $g$0 theory is canonical. On non-Witt spaces, lower and upper middle Deligne sheaves differ; no middle perverse sheaf is self-dual; and boundary conditions are required analytically (Albin et al., 2013). The later survey literature summarizes this as the canonical case versus the refined case, where analytic $g$1 Hodge theory realizes either the classical middle perversity groups or refined intermediate objects lying between the lower and upper middle theories (Albin, 2016).

2. Stratified geometry, metrics, and analytic domains

The geometric setting is a topologically stratified pseudomanifold $g$2 of dimension $g$3 with filtration

$g$4

whose strata are $g$5, with regular part $g$6 dense and distinguished neighborhoods $g$7, where $g$8 is the link of $g$9 (Albin et al., 2013). In the smooth category one works with Thom–Mather stratified spaces, equivalently with manifolds with corners carrying iterated fibration structures (Albin et al., 2013, Albin, 2016).

On a smoothly stratified pseudomanifold with an iterated incomplete edge metric $L^2$0, the analytic theory is formulated on the iie cotangent bundle $L^2$1, whose sections are locally spanned by $L^2$2 (Albin et al., 2013). The exterior derivative is initially defined on compactly supported smooth iie forms,

$L^2$3

and has minimal and maximal closed extensions

$L^2$4

$L^2$5

The reduced $L^2$6 cohomology of a closed extension $L^2$7 is

$L^2$8

These definitions are part of the standard analytic package for the conjecture (Albin et al., 2013).

On non-Witt strata, $L^2$9 is not essentially self-adjoint. Near a stratum $IH^{\bar m,*}(X)$0 with link $IH^{\bar m,*}(X)$1 and cone metric $IH^{\bar m,*}(X)$2, the orthogonal projection off $IH^{\bar m,*}(X)$3 has leading term

$IH^{\bar m,*}(X)$4

with boundary data in $IH^{\bar m,*}(X)$5. Choosing a flat subbundle

$IH^{\bar m,*}(X)$6

over the stratum defines the domain

$IH^{\bar m,*}(X)$7

Iterating this over all non-Witt strata yields an analytic mezzoperversity

$IH^{\bar m,*}(X)$8

and a closed Fredholm de Rham complex $IH^{\bar m,*}(X)$9 (Albin et al., 2013). In the operator-theoretic formulation of Cheeger spaces, the de Rham operator $L^2$0 with mezzoperversity boundary conditions is self-adjoint Fredholm with compact resolvent, and the corresponding cohomology $L^2$1 is independent of the metric (Albin et al., 2013).

3. Refined sheaf theory and mezzoperversities

The decisive refinement of the conjecture in the non-Witt case is the introduction of a category of refined middle-perversity sheaves $L^2$2 (Albin et al., 2013). These are bounded constructible complexes $L^2$3 satisfying axioms $L^2$4:

• $L^2$5 Normalization: there is an isomorphism $L^2$6 on the regular part.
• $L^2$7 Lower bound: $L^2$8 for $L^2$9, all $L^2$0.
• $L^2$1 $L^2$2-stalk vanishing: $L^2$3 for $L^2$4 and $L^2$5.
• $L^2$6 $L^2$7-costalk vanishing: $L^2$8 for $L^2$9 and $X$0.

The category $X$1 is the full subcategory of $X$2 consisting of objects satisfying these axioms. It contains the classical Deligne sheaves $X$3 and $X$4, and in general many more objects on non-Witt spaces (Albin et al., 2013).

The classification is by topological mezzoperversities. A mezzoperversity $X$5 is a collection of compatible choices of locally constant subsheaves

$X$6

at strata of odd codimension, assembled inductively via a modified truncation functor to a Deligne-type object $X$7 (Albin et al., 2013). The classification theorem states that every $X$8 arises, up to isomorphism extending the normalization on $X$9, as $g$0 for a unique mezzoperversity $g$1 (Albin et al., 2013).

The modified truncation is the mechanism by which the sheaf-theoretic theory retains the critical middle-degree data. Given an injective map of sheaves $g$2, Banagl’s modified truncation is

$g$3

with cohomology

$g$4

Inductively applying this at the relevant odd-codimension strata produces $g$5 (Albin et al., 2013).

This sheaf-theoretic refinement is the precise topological counterpart to the analytic boundary conditions. A plausible implication is that the classical conjecture is not merely extended by allowing more singular spaces; rather, its target theory is enlarged so that the analytic ambiguity at non-Witt strata is represented exactly by extra sheaf data.

4. Analytic–topological equivalence on non-Witt spaces

The central theorem of the refined theory is the equivalence between the sheaf-theoretic and analytic constructions on Thom–Mather spaces with suitably scaled iie metrics (Albin et al., 2013). Let $g$6 denote the sheafification of the presheaf $g$7. Then:

• Theorem 5.6: if $g$8 is a smoothly stratified pseudomanifold with a suitably scaled iie metric and $g$9 is an analytic mezzoperversity, then the sheaf complex $L^2$0 lies in $L^2$1.
• Theorem 5.7: on a compact smoothly stratified pseudomanifold $L^2$2, every topological mezzoperversity $L^2$3 corresponds to an analytic mezzoperversity $L^2$4, and

$L^2$5

In particular, for Witt spaces, the refined data are vacuous, $L^2$6, and one recovers the classical isomorphism

$L^2$7

(Albin et al., 2013).

The local computation underlying this bridge is the generalized Poincaré lemma. For a distinguished neighborhood $L^2$8 and a flat trivialization of $L^2$9,

$X$00

This identifies the middle-degree local cohomology sheaf with the sheaf of flat sections of $X$01, thereby realizing topological and analytic mezzoperversities as the same data (Albin et al., 2013).

The survey treatment describes this as the full resolution of the CGM correspondence for Thom–Mather pseudomanifolds: for any mezzoperversity $X$02, there is a corresponding topological mezzoperversity $X$03 and a refined Deligne-type sheaf $X$04 with

$X$05

while the extreme choices recover the lower and upper middle perversity groups (Albin, 2016). This provides the modern form of the conjecture: not simply $X$06 cohomology versus $X$07, but a dictionary between analytic domains and refined sheaf-theoretic intersection theories.

5. Duality, self-duality, Cheeger spaces, and signatures

Verdier duality preserves the refined category: $X$08 and the dual mezzoperversity $X$09 is defined as the mezzoperversity of $X$10 (Albin et al., 2013). Consequently there is a natural non-degenerate pairing

$X$11

When $X$12 is self-dual, one obtains a self-dual sheaf and generalized Poincaré duality (Albin et al., 2013).

In Banagl’s framework, Theorem 4.1 shows that self-dual sheaves lie in $X$13 after shift by $X$14, and their mezzoperversities coincide with the Lagrangian structures (Albin et al., 2013). In the smooth setting, Proposition 4.3 states that a smoothly stratified pseudomanifold is a Cheeger space if and only if it is an L-space (Albin et al., 2013). The same equivalence is presented in the signature-theoretic literature: a Cheeger space is a stratified pseudomanifold admitting a self-dual mezzoperversity, so that the $X$15 intersection pairing

$X$16

is nondegenerate and defines an analytic signature (Albin et al., 2013).

The analytic signature package is then developed in full. The signature operator with self-dual mezzoperversity is self-adjoint Fredholm, and its index equals the signature of the pairing on $X$17 (Albin et al., 2013). The theory is invariant under stratified homotopy equivalence and under Cheeger space cobordism (Albin et al., 2013). Moreover, the analytic signature equals Banagl’s topological signature, and the analytic and topological $X$18-classes agree (Albin et al., 2013).

A common misconception is that the conjecture concerns only cohomology groups. In the refined non-Witt setting, the mature theory is inseparable from duality, signatures, and self-dual structures. The cohomological identification is one component of a broader analytic–topological package.

6. Examples, variants, and developments in complex-analytic settings

The basic model is the cone $X$19. If $X$20 has nontrivial middle-degree cohomology $X$21, then $X$22 is non-Witt. Choosing $X$23 defines $X$24, and the local computation gives

$X$25

The corresponding refined intersection hypercohomology of $X$26 matches this analytic computation (Albin et al., 2013). This cone calculation also appears in later stratified de Rham models, where the critical asymptotics are described by indicial roots and the boundary trace $X$27 is projected to the chosen Lagrangian $X$28 (Luo et al., 1 May 2025).

The suspension examples illustrate obstructions to self-duality. The suspension of $X$29, $X$30, is an L-space and hence admits self-dual structures, whereas $X$31 is not, showing that not all non-Witt spaces admit self-dual refinements (Albin et al., 2013). This suggests that the existence of a cohomology theory of CGM type is broader than the existence of a signature theory.

Several later works extend the conjectural or proven picture in complex-analytic directions. One line shows that over an arbitrary compact complex space $X$32, there exists a complete Hermitian metric $X$33 on $X$34 such that

$X$35

and, when $X$36 is Kähler, $X$37 may be chosen Kähler (Shentu et al., 2021). That construction gives a fine $X$38 resolution of the intersection complex and extends to pure Hodge modules with strict supports (Shentu et al., 2021). In a more specialized setting, for a compact complex projective variety with isolated singularities, the identification

$X$39

is attributed to Ohsawa’s theorem, and later work is described as closing a technical gap by proving strong convergence of harmonic forms and limit-domain identification near the singular set (Luo et al., 3 Aug 2025).

Other papers develop stratified de Rham complexes meant to model the non-Witt refinement directly. One such framework states that for a compact Whitney stratified pseudomanifold,

$X$40

and, under an adapted iterated conic metric,

$X$41

(Luo et al., 1 May 2025). This suggests a continuing effort to recast the CGM correspondence in increasingly sheaf-theoretic or derived language while retaining the mezzoperversity mechanism.

7. Scope, limitations, and conceptual status

The status of the conjecture is therefore layered rather than uniform. For Witt spaces, the identification between $X$42 de Rham cohomology and middle perversity intersection cohomology is canonical and established (Albin et al., 2013). For non-Witt Thom–Mather spaces with suitably scaled iie metrics, the refined version is established: analytic $X$43 de Rham cohomology with ideal boundary conditions indexed by mezzoperversities is canonically isomorphic to the hypercohomology of refined middle-perversity-compatible sheaves $X$44 (Albin et al., 2013). For self-dual mezzoperversities, this yields the Cheeger-space signature theory and the Novikov package (Albin et al., 2013).

At the same time, the hypotheses remain significant. On the analytic side, the refined equivalence requires a Thom–Mather stratification and a suitably scaled iie metric, whereas the sheaf-theoretic side requires only a topological stratification (Albin et al., 2013). Mezzoperversities are not unique on non-Witt spaces; the classification theorem shows that all refined theories arise this way (Albin et al., 2013). Existence of self-dual mezzoperversities is obstructed, for example by link signatures; L-spaces may fail to exist (Albin et al., 2013).

A further limitation is that the refined theory developed in (Albin et al., 2013) is middle-perversity-compatible; extension to other perversities is noted as possible but not pursued there. In the complex-analytic literature, the existence of suitable complete Hermitian metrics can depend strongly on the construction, and some works explicitly state that metrics such as the raw Fubini–Study restriction are not covered by their methods (Shentu et al., 2021). This suggests that while the conjecture is resolved in the Thom–Mather/iie framework and in several important complex-analytic settings, metric-specific formulations may still require separate analysis.

In contemporary usage, the term “Cheeger–Goresky–MacPherson conjecture” therefore denotes both the classical Witt-space identification and its refined non-Witt generalization. The mature form of the subject is the analytic–topological equivalence: $X$45 with the classical identity

$X$46

recovered as the special case in which the refined data are vacuous (Albin et al., 2013).