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

The paper establishes that the diffeological de Rham cohomology of the quotient space $G/H$, where $H$ is a dense subgroup of a Lie group $G$, is isomorphic to the Lie algebra cohomology of the quotient $\mathfrak g/\mathfrak h$, with $\mathfrak h$ defined as the ideal of elements in $\mathfrak g$ whose one-parameter subgroups lie entirely within $H$.

The Gist

Here is an explanation of the paper "The de Rham Cohomology of a Lie Group Modulo a Dense Subgroup" using simple language and creative analogies.

The Big Picture: What are they trying to measure?

Imagine you have a smooth, perfect sphere (this is your Lie Group, let's call it $G$). Now, imagine you have a very strange, invisible net made of tiny, infinitely dense threads wrapped around this sphere (this is your subgroup, $H$).

In the old days of math, if you tried to look at the "shape" of the sphere after you squashed it down by these threads (creating the quotient space $G/H$), you would get a mess. Because the threads are so dense, they touch every single point on the sphere. If you tried to measure the distance between two points on the resulting shape, you'd find they are actually zero distance apart. The shape collapses into a single, featureless blob. In standard topology, this shape is "trivial"—it has no holes, no bumps, nothing. It's just a point.

The Problem: If the shape is just a point, it has no "cohomology" (a fancy math word for counting holes, loops, and voids). But the authors, Brant Clark and François Ziegler, say: "Wait a minute. Just because the shape looks like a point to our eyes doesn't mean it doesn't have a hidden structure."

They want to find a way to measure the "holes" in this collapsed shape that standard math misses.

The Solution: A New Pair of Glasses (Diffeology)

The authors use a new mathematical framework called Diffeology. Think of this as putting on a special pair of glasses that doesn't look at distance (topology), but looks at smoothness and paths.

• Standard View: "Can I walk from point A to point B without jumping?" (If the threads are dense, the answer is "No, you can't distinguish them.")
• Diffeological View: "Can I draw a smooth line through this space?" (Even if the threads are everywhere, can we still trace a smooth path through the weave?)

Using these glasses, the authors prove a surprising theorem: The "holes" in this collapsed, dense shape are exactly the same as the "holes" in a specific algebraic recipe derived from the Lie Group's internal structure.

The Analogy: The Infinite Weave

Let's break down the main result (Theorem 0.2) with a metaphor.

1. The Lie Group ($G$) is a Giant Factory.
Inside this factory, there are machines (the Lie Algebra $\mathfrak{g}$) that generate all the movement. The factory is smooth and continuous.

2. The Dense Subgroup ($H$) is a Secret Society.
This society is so active that its members are everywhere in the factory. You can't find a spot in the factory where a member isn't standing.

• The Catch: Even though they are everywhere, they aren't random. They follow a strict, hidden rule. Some of them move in straight lines (like a train on a track), others might move in a spiral.

3. The Quotient ($G/H$) is the "Shadow" of the Factory.
If you squish the factory down by identifying everyone in the Secret Society as "the same person," the factory collapses. To a normal eye, it's a single point.

4. The Magic Trick:
The authors show that you don't need to look at the collapsed factory to understand its shape. Instead, you just need to look at the blueprint of the machines that the Secret Society doesn't use.

• They define a special list of "forbidden moves" (the ideal $\mathfrak{h}$).
• They look at the "allowed moves" (the difference between the factory machines and the forbidden moves, written as $\mathfrak{g}/\mathfrak{h}$).
• The Result: The number of holes in the collapsed factory is exactly the same as the number of holes you can find in the algebra of the "allowed moves."

Why is this cool? (The "D-Connected" vs. "D-Discrete" Cases)

The paper explores two extreme scenarios to show how flexible this new method is:

Scenario A: The "Irrational Winding" (The Quasi-Circle)
Imagine a 2D torus (a donut). You draw a line on it that winds around forever, never repeating its path, getting closer and closer to every point on the donut (like a irrational slope on a grid).

• Old Math: The line is so dense it covers the whole donut. The "space" left over is a point.
• New Math: The authors say, "Actually, this space behaves exactly like a circle!"
• Why? Because the "allowed moves" in the algebra correspond to a circle. So, even though the shape looks like a point, its "smoothness structure" remembers that it used to be a circle. It has 1 hole.

Scenario B: The "D-Discrete" Case (The Dust)
Imagine the Secret Society is made of individual, isolated points scattered everywhere (like dust).

• Old Math: The space is a mess of disconnected points.
• New Math: The authors show that the "holes" in this space are exactly the same as the holes in the original Lie Group's algebra.
• The Punchline: This means you can take any Lie group (like a sphere, a torus, or a complex 3D shape), find a dense cloud of points inside it, and the "smooth shape" of the remaining space will perfectly mimic the algebraic complexity of the original group.

The "So What?"

This paper is a bridge between two worlds that usually don't talk to each other:

1. Geometry: The study of shapes and spaces.
2. Algebra: The study of equations and abstract structures.

Usually, if a geometric space collapses (because a subgroup is dense), the geometry is lost. The authors say: "No, the geometry isn't lost; it just moved into the algebra."

They provide a dictionary. If you want to know the shape of a "collapsed" space, you don't need to struggle with the messy geometry. You just need to do a simple algebra calculation (Lie Algebra Cohomology) on the "difference" between the group and the subgroup.

Summary in One Sentence

Even when a dense subgroup crushes a Lie group into a shapeless blob, the "smooth" structure of that blob is secretly preserved and can be perfectly calculated using a simple algebraic formula based on the group's internal machinery.

Technical Summary

Here is a detailed technical summary of the paper "The de Rham Cohomology of a Lie Group Modulo a Dense Subgroup" by Brant Clark and François Ziegler.

1. Problem Statement

The paper addresses a fundamental gap in the differential topology of Lie groups when dealing with non-closed subgroups.

• The Issue: If $H$ is a non-closed (specifically dense) subgroup of a Lie group $G$, the standard quotient space $G/H$ equipped with the quotient topology is trivial (indiscrete) and not Hausdorff. Consequently, the standard algebra of smooth functions on $G/H$ is trivial, rendering classical de Rham cohomology undefined or useless.
• The Goal: To define a meaningful de Rham cohomology for the quotient space $G/H$ where $H$ is a dense subgroup, and to compute it explicitly in terms of Lie algebra data.
• Context: Previous approaches (e.g., Connes' non-commutative geometry) attempted to replace the trivial function algebra with crossed product algebras and cyclic cohomology. The authors instead utilize diffeological spaces to retain a commutative geometric framework.

2. Methodology

The authors employ the framework of diffeological spaces, a generalization of smooth manifolds introduced by Souriau.

• Diffeological Structure:

  • Instead of charts, a diffeological space is defined by a set of "plots" (smooth maps from Euclidean open sets into the space) satisfying covering, locality, and smooth compatibility axioms.
  • Quotient Diffeology: For $X = G/H$, the authors endow it with the quotient diffeology. This ensures that a map out of $X$ is smooth if and only if its composition with the projection $\pi: G \to G/H$ is smooth.
  • Diffeological Forms: A $k$-form on a diffeological space is defined as a collection of ordinary forms on the domain of every plot, compatible with smooth reparametrizations. This allows the construction of a de Rham complex $(\Omega^\bullet(G/H), d)$ even when the topology is trivial.

• Key Technical Steps:

  1. Characterization of Forms: The authors prove that the pullback via the projection $\pi: G \to G/H$ induces a bijection between diffeological forms on $G/H$ and specific forms on $G$.
  2. Invariance and Horizontality: They show that forms on $G/H$ correspond exactly to forms on $G$ that are:
     • Right-invariant: Invariant under the action of $G$ (due to the density of $H$).
     • $\mathfrak{h}$-horizontal: Vanishing on vectors in the Lie algebra $\mathfrak{h}$.
  3. Lie Algebra Identification: By switching to left-invariant forms via the inversion map, the problem reduces to the Chevalley–Eilenberg complex of the Lie algebra.
  4. Ideal Property: A crucial lemma establishes that if $H$ is dense, the associated Lie algebra $\mathfrak{h} = \{Z \in \mathfrak{g} : \exp(tZ) \in H, \forall t \in \mathbb{R}\}$ is an ideal in $\mathfrak{g}$. This allows the reduction to the quotient algebra $\mathfrak{g}/\mathfrak{h}$.

3. Key Contributions and Results

Main Theorem (Theorem 0.2)

Let $H$ be a dense subgroup of a Lie group $G$ with Lie algebra $\mathfrak{g}$. Let $\mathfrak{h}$ be the ideal defined above. Then there is an isomorphism of complexes:
$$(\Omega^\bullet(G/H), d) \cong (\Lambda^\bullet(\mathfrak{g}/\mathfrak{h})^*, d)$$
Consequently, the diffeological de Rham cohomology is isomorphic to the Lie algebra cohomology of the quotient:
$$H^\bullet_{dR}(G/H) \cong H^\bullet(\mathfrak{g}/\mathfrak{h})$$
Significance: This result is "doubly unusual":

1. Manifolds typically have infinite-dimensional cohomology rings; here, the cohomology is finite-dimensional (determined by the dimension of $\mathfrak{g}/\mathfrak{h}$).
2. Despite the quotient having a trivial topology, the cohomology is non-trivial.

Corollaries and Special Cases (Corollary 0.4)

The paper analyzes specific structural cases of $H$:

• Case (a) D-connected $H$: If $H$ is connected in the diffeological sense (manifold topology), then $\mathfrak{g}/\mathfrak{h}$ is abelian. The cohomology is the full exterior algebra $\Lambda^\bullet(\mathfrak{g}/\mathfrak{h})^*$.
  • Example: The irrational winding on a torus ($G=T^2, H \cong \mathbb{R}$). The quotient is a "quasicircle," and its cohomology is $\mathbb{R} \oplus \mathbb{R}$, identical to a standard circle.
• Case (b) D-discrete $H$: If $H$ is discrete in the diffeological sense, then $\mathfrak{h} = \{0\}$. The cohomology is $H^\bullet(\mathfrak{g})$. This implies that any Lie algebra cohomology ring can be realized as the de Rham cohomology of a dense quotient.
• Case (c) Vector Space Quotients: For $V/A$ where $A$ is a dense, discrete additive subgroup, the cohomology is the full exterior algebra $\Lambda^\bullet V^*$.

Comparison with Other Theories

The authors contrast their results with:

• Closed Subgroups: For closed $H$, the result relates to relative Lie algebra cohomology $H^\bullet(\mathfrak{g}, \mathfrak{h})$.
• Non-commutative Geometry: They note that for the irrational torus, their result ($\mathbb{R} \oplus \mathbb{R}$) differs from the periodic cyclic cohomology of the associated crossed product algebra (which yields $\mathbb{R} \oplus \mathbb{R}^2 \oplus \mathbb{R}$). The diffeological approach captures the "geometric" intuition of the space as a "quasi-circle" rather than a non-commutative algebraic object.
• Čech Cohomology: The diffeological de Rham cohomology differs from the diffeological Čech cohomology (which recovers the group cohomology of $A$).

4. Significance and Implications

• Resolution of the "Trivial Topology" Paradox: The paper demonstrates that the lack of a Hausdorff topology does not preclude the existence of a rich differential geometric structure. By using diffeology, one can recover cohomological invariants that reflect the algebraic structure of the group and subgroup.
• Unification: It provides a unified framework connecting Lie group quotients, Lie algebra cohomology, and diffeological geometry.
• New Examples: The paper constructs explicit examples of spaces with "exotic" cohomology (e.g., realizing any Lie algebra cohomology ring as a quotient of a Lie group by a dense subgroup).
• Independence: The authors note that H. Kihara independently obtained the main theorem (0.2) in forthcoming work, validating the significance of the result.

5. Conclusion

Clark and Ziegler successfully extend de Rham theory to the realm of dense subgroups by utilizing diffeological spaces. They prove that the cohomology of such quotients is entirely determined by the Lie algebra of the ambient group modulo the Lie algebra of the subgroup. This bridges the gap between the algebraic properties of dense subgroups and the geometric intuition of smooth forms, offering a robust alternative to non-commutative geometric approaches for these singular spaces.