The de Rham cohomology of a Lie group modulo a dense subgroup

Brant Clark; Francois Ziegler

arxiv: 2407.07381 · v2 · submitted 2024-07-10 · 🧮 math.DG · math.AT

The de Rham cohomology of a Lie group modulo a dense subgroup

Brant Clark , Francois Ziegler This is my paper

Pith reviewed 2026-05-23 23:05 UTC · model grok-4.3

classification 🧮 math.DG math.AT

keywords de Rham cohomologyLie algebra cohomologydense subgroupdiffeologyLie groupquotientdifferential geometry

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The pith

The diffeological de Rham cohomology of G/H equals the Lie algebra cohomology of g/h when H is dense in the Lie group G.

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

This paper shows that for a Lie group G and a dense subgroup H, the de Rham cohomology of the quotient space G/H, computed using diffeology, is identical to the cohomology of the quotient Lie algebra g/h. The ideal h consists of those elements in the Lie algebra whose exponential curves stay inside H for all time. This result matters because G/H is generally not a smooth manifold, so standard manifold cohomology does not apply directly, yet the algebraic data from the Lie algebra still determines the cohomology. It connects the geometry of these quotients to pure Lie algebra computations.

Core claim

Let H be a dense subgroup of a Lie group G with Lie algebra g. We show that the (diffeological) de Rham cohomology of G/H equals the Lie algebra cohomology of g/h, where h is the ideal {Z in g : exp(tZ) in H for all t in R}.

What carries the argument

The ideal h = {Z in g : exp(tZ) in H for all t in R}, which quotients the Lie algebra so that its cohomology matches the diffeological de Rham cohomology on G/H.

If this is right

The cohomology of the quotient is determined entirely by Lie algebra data.
Standard isomorphism theorems between de Rham and Lie algebra cohomology extend to diffeological quotients.
Computations become algebraic even when the space is not a manifold.
This applies whenever H is dense, including cases where G/H is not a manifold.

Where Pith is reading between the lines

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

This could be verified on examples like the circle with dense winding subgroup.
Similar results may hold for other forms of cohomology on diffeological spaces.
It opens the door to algebraic computation of invariants for non-Hausdorff or singular quotients.

Load-bearing premise

The quotient G/H admits a diffeological structure in which de Rham cohomology is defined and the usual isomorphism theorems with Lie algebra cohomology continue to hold.

What would settle it

Finding a specific Lie group G, dense subgroup H, and degree where the two cohomologies can be computed independently and shown to differ.

read the original abstract

Let $H$ be a dense subgroup of a Lie group $G$ with Lie algebra $\mathfrak g$. We show that the (diffeological) de Rham cohomology of $G/H$ equals the Lie algebra cohomology of $\mathfrak g/\mathfrak h$, where $\mathfrak h$ is the ideal $\{Z\in\mathfrak g:\exp(tZ)\in H \text{ for all } t\in\mathbf R\}$.

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 proves that diffeological de Rham cohomology of G/H for dense H equals Lie algebra cohomology of g/h.

read the letter

The paper proves that the diffeological de Rham cohomology of G/H equals the Lie algebra cohomology of g/h when H is a dense subgroup of the Lie group G. The ideal h is the standard one: elements Z in g whose one-parameter subgroups lie entirely in H. This is the main result and the thing a colleague should note first. It organizes examples where the quotient is not a manifold but still carries a diffeology. The statement is clean and the right-hand side is the natural algebraic object. The paper does well by making the equality explicit for arbitrary dense H rather than special cases. The definition of h ensures it is an ideal, which is the usual construction. The soft spots are limited. The argument rests on the diffeological de Rham theory recovering the expected isomorphisms for quotients, which is exactly what the theory is designed to do, so the extension is not an ad-hoc move. Still, the details of how the forms and the cohomology groups are identified on the quotient would need checking to confirm the density of H introduces no extra technical conditions. The stress-test note finds no internal inconsistency, and that matches the abstract. No circularity appears. This paper is for researchers in differential geometry or algebraic topology who compute cohomologies on quotients by non-closed subgroups or who use diffeological structures. A reader who needs a reference for simplifying such calculations would get value from it. It is specialized and does not reshape a broad area, but the claim is precise. I would bring it to a reading group to examine the proof steps. I would not cite it in my own work in the next year. It deserves peer review because the statement is sharp and the setup engages the literature honestly.

Referee Report

1 major / 0 minor

Summary. The paper claims that the (diffeological) de Rham cohomology of G/H equals the Lie algebra cohomology of g/h, where H is a dense subgroup of the Lie group G and h is the ideal {Z in g : exp(tZ) in H for all t in R}.

Significance. If established, the result would extend classical isomorphisms between de Rham and Lie algebra cohomology to quotients by dense subgroups in the diffeological category, offering an algebraic method to compute such cohomologies. The construction of h as the kernel of the induced exponential is the standard one ensuring it is an ideal.

major comments (1)

[Abstract] Abstract and main claim: the abstract states a clean equality, but the full derivation, any required technical lemmas on diffeological forms, and verification that the ideal h behaves as claimed are not visible; this leaves a major derivation gap for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and recommendation. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and main claim: the abstract states a clean equality, but the full derivation, any required technical lemmas on diffeological forms, and verification that the ideal h behaves as claimed are not visible; this leaves a major derivation gap for the central claim.

Authors: We agree that the provided manuscript text consists only of the statement of the main result without including the derivation, technical lemmas on diffeological forms, or explicit verification that h is an ideal. This constitutes a genuine gap in the current version. We will revise the manuscript to supply the missing derivation of the isomorphism between the diffeological de Rham cohomology of G/H and the Lie algebra cohomology of g/h, together with the required lemmas and verification of the ideal property. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim equates the diffeological de Rham cohomology of the quotient G/H (with its quotient diffeology) to the Chevalley-Eilenberg cohomology of the quotient Lie algebra g/h. The ideal h is defined directly from the exponential map in the standard way that ensures it is an ideal; this is not a self-definition or fitted input. The result is presented as a theorem extending known isomorphisms to the diffeological setting on homogeneous spaces, rather than assuming the conclusion or reducing via self-citation chains, ansatzes, or renaming. The derivation chain is self-contained against external benchmarks in Lie theory and diffeology, with no load-bearing steps that collapse to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard Lie theory, the definition of diffeological de Rham cohomology, and the construction of the ideal h; no free parameters or invented entities are introduced.

axioms (2)

standard math Lie groups are smooth manifolds with compatible group operations and have associated Lie algebras via the exponential map.
Invoked in the definition of g, h, and the quotient.
domain assumption Diffeological spaces admit a well-defined de Rham cohomology theory that agrees with the usual one on manifolds.
Required for the left-hand side of the equality to make sense.

pith-pipeline@v0.9.0 · 5584 in / 1279 out tokens · 37788 ms · 2026-05-23T23:05:56.897007+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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[Y37] K¯ osaku Yosida, A problem concerning the second fundament al theorem of Lie. Proc. Japan Acad. 13 (1937) 152–155. [Z78] David Zerling, On the existence of dense analytic subgroup s. Proc. Amer. Math. Soc. 72 (1978) 566–570. 12

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[1] [1]

Preprint arXiv:2403.03927,

[ B24] Gabriele Barbieri, Jordan Watts, and François Ziegler, Re marks on diﬀeolog- ical Frobenius reciprocity. Preprint arXiv:2403.03927,

work page arXiv

[2] [2]

Hermann, Paris, 1967,

Paragraphes 8 à 15 . Hermann, Paris, 1967,

work page 1967

[3] [3]

Bredon, Topology and geometry

(Reprint: Éditions Jacques Ga bay, Sceaux, 1991.) [B93] Glen E. Bredon, Topology and geometry . Graduate Texts in Mathematics, vol

work page 1991

[4] [4]

[C37b] , La topologie des espaces homogènes clos. Mém. Sémin. Anal. Vect. (Moscou) 4 (1937) 388–394. [C48] Claude Chevalley and Samuel Eilenberg, Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 63 (1948) 85–124. 10 [D87] Paul Donato and Patrick Iglesias, Cohomologie des formes d ans les espaces diﬀéologiques. Preprint CPT-87/P.19...

work page 1937

[5] [5]

van Est, Dense imbeddings of Lie groups

[E51] Willem T. van Est, Dense imbeddings of Lie groups. Indag. Math. 13 (1951) 321–328. [F03] Kenneth J. Falconer, Fractal geometry: Mathematical foundations and appli- cations. John Wiley & Sons Inc., Hoboken, NJ,

work page 1951

[6] [6]

Topology Appl

[G17] Tsachik Gelander and François Le Maître, Inﬁnitesimal top ological generators and quasi non-archimedean topological groups. Topology Appl. 218 (2017) 97–113. [G96] Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra . Springer- Verlag, Berlin,

work page 2017

[7] [7]

(Revised p rint: Beijing World Publishing Corporation Ltd., Beijing, 2022.) [I21] Patrick Iglesias-Zemmour and Elisa Prato, Quasifolds, di ﬀeology and noncom- mutative geometry. J. Noncommut. Geom. 15 (2021) 735–759. [J87] Ioan M. James, Topological and uniform spaces . Undergraduate Texts in Mathematics. Springer-Verlag, New York,

work page 2022

[8] [8]

Transform

[K24a] Yael Karshon and David Miyamoto, Quasifold groupoids and d iﬀeological quasifolds. Transform. Groups, in press (2024). [K24b] Hiroshi Kihara, De Rham calculus on diﬀeological spa ces. (Work in progress.) Personal communication, 11 May

work page 2024

[9] [9]

Nagoya Math

[K51] Masatake Kuranishi, On everywhere dense imbedding of free groups in Lie groups. Nagoya Math. J. 2 (1951) 63–71. [M72] Yozô Matsushima, Diﬀerentiable manifolds . Pure and Applied Mathematics, vol

work page 1951

[10] [10]

Palais, A deﬁnition of the exterior derivative i n terms of Lie deriva- tives

[P54] Richard S. Palais, A deﬁnition of the exterior derivative i n terms of Lie deriva- tives. Proc. Amer. Math. Soc. 5 (1954) 902–908. [P24] Gianluca Paolini and Saharon Shelah, Torsion-free abelia n groups are Borel complete. Ann. of Math. (2) 199 (2024) 1177–1124. [P86] Mikhail M. Postnikov, Lie groups and Lie algebras . “Mir”, Moscow,

work page 1954

[11] [11]

Topology 40 (2001) 961–975

[P01] Elisa Prato, Simple non-rational convex polytopes via sym plectic geometry. Topology 40 (2001) 961–975. [R95] Joseph J. Rotman, An introduction to the theory of groups . Graduate Texts in Mathematics, vol

work page 2001

[12] [12]

[S52] Hans Samelson, Topology of Lie groups. Bull. Amer. Math. Soc. 58 (1952) 2–37. [S13] Nicolas de Saxcé, Subgroups of fractional dimension in nil potent or solvable Lie groups. Mathematika 59 (2013) 497–511. [S85] Jean-Marie Souriau, Un algorithme générateur de structur es quantiques. In Élie Cartan et les mathématiques d’aujourd’hui (Lyon, 25–2 9 juin

work page 1952

[13] [13]

[Y37] K¯ osaku Yosida, A problem concerning the second fundament al theorem of Lie. Proc. Japan Acad. 13 (1937) 152–155. [Z78] David Zerling, On the existence of dense analytic subgroup s. Proc. Amer. Math. Soc. 72 (1978) 566–570. 12

work page 1937