Persistent de Rham Cohomology

• Persistent de Rham Cohomology is a framework that studies de Rham cohomology extensions over filtrations, singular spaces, and nilpotent thickenings.
• It employs models like Vietoris–Rips filtrations, equivariant cohomology, and relative de Rham theorems to capture persistent invariants.
• The approach emphasizes functorial persistence, stability, and multiplicative structures to compare graded cohomology across different geometries.

Searching arXiv for recent and relevant papers on persistent de Rham cohomology and adjacent formulations. Searching arXiv for "persistent de Rham cohomology". Searching arXiv for persistent cohomology operations and equivariant cohomology. Searching arXiv for deformation-theoretic de Rham persistence in characteristic $p$. Persistent de Rham cohomology is best understood as a family of adjacent research programs rather than a single standardized construction. The literature surveyed here suggests at least four distinct but related meanings. In topological data analysis, one seeks de Rham-type models for filtrations $M_r\hookrightarrow M_{r'}$ and the induced contravariant systems in cohomology, often through Vietoris–Rips or neighborhood filtrations; however, the most relevant papers develop persistent cohomology operations and persistent Borel equivariant cohomology rather than a persistent de Rham complex itself (Medina-Mardones et al., 21 Mar 2025, Adams et al., 2024). In singular geometry, $L^2$ de Rham theory, intersection cohomology, and multiplicative relative de Rham theorems provide models that could support persistence-like constructions on stratified spaces (Bei, 2011, Schlöder et al., 2019). In arithmetic geometry, the term points to a different phenomenon entirely: de Rham cohomology over $\mathbb F_p$ persists uniquely across Artinian nilpotent thickenings, so that crystalline cohomology is its unique functorial deformation (Mondal, 2021).

1. Terminological scope and principal interpretations

The available literature suggests that the expression “persistent de Rham cohomology” does not designate a single canonical theory. One strand is genuinely persistent in the filtration sense: a filtered space is treated as a functor

$(R,\le)\longrightarrow \mathrm{Top},$

and one studies the resulting diagrams in cohomology or refinements thereof. Another strand is equivariant, where the filtration lives in $G\text{-}\mathrm{Top}$ and the relevant groups are Borel equivariant cohomology groups $H_G^*(Y_r)$. A third strand is singular and analytic, where weighted $L^2$ de Rham theory on stratified pseudomanifolds is compared with intersection cohomology and varies with metric or perversity data. A fourth strand is deformation-theoretic: persistence means unique extension of a cohomology functor across infinitesimal thickenings (Medina-Mardones et al., 21 Mar 2025, Adams et al., 2024, Bei, 2011, Mondal, 2021).

A common misconception is to identify all uses of “persistence” with barcode-type persistence modules. Mondal’s “$G_a^{\mathrm{perf}}$-modules and de Rham Cohomology” explicitly concerns deformation-rigidity of algebraic de Rham cohomology in characteristic $p$, not persistence in the topological data analysis sense. There are no barcodes, persistence modules over $M_r\hookrightarrow M_{r'}$0 or $M_r\hookrightarrow M_{r'}$1, Vietoris–Rips or Čech filtrations, or persistent homology constructions in that work (Mondal, 2021).

2. Filtered spaces, Vietoris–Rips constructions, and de Rham-style models

In the topological persistence framework of persistent cohomology operations, an $M_r\hookrightarrow M_{r'}$2-space is a functor

$M_r\hookrightarrow M_{r'}$3

with structure maps $M_r\hookrightarrow M_{r'}$4 for $M_r\hookrightarrow M_{r'}$5. Persistent homology is written $M_r\hookrightarrow M_{r'}$6, while persistent cohomology is contravariant in the filtration parameter and is treated as a functor $M_r\hookrightarrow M_{r'}$7. The main metric filtration is the Vietoris–Rips filtration

$M_r\hookrightarrow M_{r'}$8

so that $M_r\hookrightarrow M_{r'}$9 for $L^2$0 (Medina-Mardones et al., 21 Mar 2025).

This framework is directly relevant to any de Rham version because the same formal input would be required: a filtration by spaces, functorial maps induced by inclusions, and a contravariant cohomology theory. The paper on persistent cohomology operations uses neighborhood filtrations

$L^2$1

for compact metric spaces and cites the natural homotopy equivalence

$L^2$2

For smooth closed Riemannian manifolds $L^2$3, one may think of $L^2$4 as the geometric thickening whose cohomology could, in principle, be studied by differential forms if a suitable smooth model is chosen. This suggests a route to persistent de Rham cohomology: replace $L^2$5 by a de Rham model $L^2$6 when smoothness is available, and then study the induced maps along the filtration (Medina-Mardones et al., 21 Mar 2025).

Persistent equivariant cohomology uses the same categorical structure. A filtration of spaces $L^2$7 is a functor $L^2$8, while a filtration of $L^2$9-spaces is a functor $\mathbb F_p$0 with $\mathbb F_p$1-equivariant structure maps. Applying Borel equivariant cohomology produces a contravariant system

$\mathbb F_p$2

For de Rham purposes, this matters because the formal persistence mechanism is already present at the level of filtrations and functorial pullbacks, even though the paper itself works with singular/Borel cohomology rather than differential forms (Adams et al., 2024).

3. Persistent operations and equivariant refinements

Persistent cohomology operations enlarge the information retained by a persistence module. For a linear cohomology operation

$\mathbb F_p$3

the induced persistent invariants are

$\mathbb F_p$4

These are persistent cohomology operation modules, and their barcodes are the $\mathbb F_p$5-barcode and $\mathbb F_p$6-barcode. The theory strictly generalizes ordinary persistent cohomology because

$\mathbb F_p$7

For cellular $\mathbb F_p$8-spaces $\mathbb F_p$9, the stability theorem gives

$(R,\le)\longrightarrow \mathrm{Top},$0

and for metric spaces,

$(R,\le)\longrightarrow \mathrm{Top},$1

The paper then constructs pairs of Riemannian pseudomanifolds for which the Gromov–Hausdorff estimates derived from persistent cohomology operations are strictly sharper than those obtained using persistent homology; the explicit comparison $(R,\le)\longrightarrow \mathrm{Top},$2 versus a wedge of spheres is achieved using Steenrod squares and the long bar in $(R,\le)\longrightarrow \mathrm{Top},$3 that ordinary homology does not detect (Medina-Mardones et al., 21 Mar 2025).

For persistent de Rham cohomology, the significance is methodological rather than literal. The paper does not explicitly use de Rham complexes, differential forms, or de Rham cohomology models, and there is no filtered dg-algebra of forms, no persistent de Rham theorem, and no chain-level de Rham representative for Steenrod operations. Still, it suggests that a de Rham-persistent theory should not stop at graded vector spaces and should instead seek stable invariants extracted from richer filtered algebraic structures such as cup products, Massey products, or minimal-model data (Medina-Mardones et al., 21 Mar 2025).

Persistent equivariant cohomology sharpens this point by showing how much additional structure becomes visible when symmetry is retained. Borel equivariant cohomology is defined by

$(R,\le)\longrightarrow \mathrm{Top},$4

and for the circle action on Vietoris–Rips metric thickenings of $(R,\le)\longrightarrow \mathrm{Top},$5, the main theorem states that if

$(R,\le)\longrightarrow \mathrm{Top},$6

then

$(R,\le)\longrightarrow \mathrm{Top},$7

With coefficients $(R,\le)\longrightarrow \mathrm{Top},$8, this simplifies to

$(R,\le)\longrightarrow \mathrm{Top},$9

The paper is explicit that the integral formula detects torsion and fixed-point information that ordinary de Rham cohomology over $G\text{-}\mathrm{Top}$0 would not see. It also suggests a de Rham analogue via the Cartan complex

$G\text{-}\mathrm{Top}$1

for compact Lie groups acting smoothly, while not constructing such a theory (Adams et al., 2024).

4. Relative, singular, and multiplicative de Rham models

One of the strongest direct bridges to persistent de Rham cohomology is the multiplicative relative de Rham theorem for pairs of smooth manifolds. For a pair $G\text{-}\mathrm{Top}$2, relative de Rham cohomology is defined through the kernel of restriction,

$G\text{-}\mathrm{Top}$3

and the paper proves a multiplicative isomorphism between relative de Rham and relative singular cohomology. It also proves a cellular multiplicative relative de Rham theorem

$G\text{-}\mathrm{Top}$4

and, for stratified pseudomanifolds with isolated singularities, a multiplicative de Rham theorem for intersection space cohomology: $G\text{-}\mathrm{Top}$5 as cohomology rings. The central point is that the comparison preserves wedge and cup products, not only graded vector-space structures (Schlöder et al., 2019).

For singular spaces, Bei’s $G\text{-}\mathrm{Top}$6 de Rham and Hodge theory gives a second major model. On the regular part of a compact smoothly stratified pseudomanifold, equipped with a quasi edge metric with weights,

$G\text{-}\mathrm{Top}$7

the metric determines perversities $G\text{-}\mathrm{Top}$8 and $G\text{-}\mathrm{Top}$9. The main global identifications are

$H_G^*(Y_r)$0

and

$H_G^*(Y_r)$1

Local cone analysis on

$H_G^*(Y_r)$2

controls when $H_G^*(Y_r)$3 cohomology survives or vanishes and thereby explains how the metric data determine the allowable singular behavior (Bei, 2011).

These two papers do not construct persistence modules, but they provide much of the infrastructure that such a theory would require. The singular $H_G^*(Y_r)$4 theory suggests a piecewise constant family of de Rham-type invariants indexed by metric weights $H_G^*(Y_r)$5, while the relative and multiplicative theorems show that de Rham-level and topological-level structures can agree ring-theoretically on pairs and on certain singular spaces. A plausible implication is that any serious singular persistent de Rham theory will need both ingredients: analytic control near singular strata and multiplicative compatibility across inclusions or pairs.

5. Deformation-theoretic persistence in characteristic $H_G^*(Y_r)$6

In arithmetic geometry, persistence takes a different form. Algebraic de Rham cohomology

$H_G^*(Y_r)$7

is proved to be formally étale in the precise sense that for any Artinian local ring $H_G^*(Y_r)$8 with residue field $H_G^*(Y_r)$9, the functor $L^2$0 admits a deformation

$L^2$1

together with an identification

$L^2$2

and the $L^2$3-groupoid of such deformations is contractible. Equivalently, all infinitesimal deformations exist uniquely (Mondal, 2021).

This is not persistence in the barcode sense, and the paper is explicit on that point. Its persistence phenomenon is rigidity across infinitesimal thickenings: de Rham cohomology over $L^2$4 determines all its Artinian liftings. For $L^2$5, crystalline cohomology provides such a deformation, so crystalline cohomology is the unique functorial deformation of de Rham cohomology. The mechanism passes through pointed $L^2$6-modules, quasi-ideals in Drinfeld’s sense, and the unwinding construction. On quasiregular semiperfect algebras, the decisive identifications are

$L^2$7

with $L^2$8. In this sense, de Rham cohomology persists canonically through nilpotent extensions of the base ring (Mondal, 2021).

6. Structural limitations and emerging directions

The current literature is marked as much by what it does not yet provide as by what it does. The persistent cohomology-operations paper gives a stability architecture, decomposition formulas, and Gromov–Hausdorff applications, but it does not construct a persistent de Rham complex, a filtered de Rham theorem, or a chain-level differential-form realization of the operations (Medina-Mardones et al., 21 Mar 2025). The persistent equivariant cohomology paper defines persistent Borel equivariant cohomology and computes it explicitly for circle actions on Vietoris–Rips metric thickenings, but it does not provide barcode decompositions and its most striking formulas are integral and torsion-sensitive, beyond what ordinary real de Rham theory can detect (Adams et al., 2024).

The singular and relative papers are likewise preparatory rather than definitive from a persistence standpoint. The $L^2$9 de Rham and Hodge theorem for stratified pseudomanifolds does not define filtered objects, interleavings, or canonical maps between different metric or perversity choices, even though it identifies the groups that such a persistence theory might track (Bei, 2011). The multiplicative de Rham theorem for relative and intersection space cohomology proves ring-level comparisons levelwise, but it does not develop persistence modules, filtered de Rham complexes, or functoriality of intersection spaces across inclusions (Schlöder et al., 2019).

Taken together, these works suggest a coherent research agenda. A persistent de Rham theory in the topological sense would likely need a filtered dg-algebra or relative-form model compatible with Vietoris–Rips or neighborhood filtrations, a way to retain multiplicative or equivariant information, and a singular extension accommodating stratified pseudomanifolds. A persistent de Rham theory in the arithmetic sense is already realized as infinitesimal persistence: unique deformation of de Rham cohomology to crystalline cohomology across Artinian nilpotent thickenings. The term therefore names not one object but a convergent landscape of theories in which de Rham-type cohomology is required to vary functorially, stably, and, in the strongest formulations, multiplicatively across a parameter (Medina-Mardones et al., 21 Mar 2025, Adams et al., 2024, Bei, 2011, Schlöder et al., 2019, Mondal, 2021).