A Transverse Averaging Operator and Cohomology of Quotients by Non-closed Subgroups

Abstract: In this article, we introduce a transverse averaging operator for basic forms on a Riemannian foliation equipped with an isometric transverse Lie algebra action, under the assumption that the leaf closure space is compact. Unlike the classical averaging operator in equivariant geometry, which is defined by integration over a compact Lie group, our operator is built purely from infinitesimal transverse data and does not require any global group action. We show that it sends every closed basic form to an invariant basic form representing the same basic cohomology class. As a main application, we compute the diffeological de Rham cohomology of the homogeneous space $G/H$, where $G$ is a connected Lie group, not necessarily compact, and $H$ is a connected Lie subgroup, not necessarily closed. Let $\mathfrak g$ and $\mathfrak h$ be the Lie algebras of $G$ and $H$, respectively. Assuming that $\mathfrak g$ is of compact type and that $G/\overline{H}$ is compact, we prove that [ H^\bullet_{dR}(G/H)\cong H^\bullet(\mathfrak g,\mathfrak h). ] If, in addition, $\mathfrak h$ is an ideal in $\mathfrak g$, then under the weaker assumption that $G/\overline{H}$ is compact, we obtain [ H^\bullet_{dR}(G/H)\cong H^\bullet(\mathfrak g/\mathfrak h), ] without requiring $\mathfrak g$ to be of compact type.

• The paper introduces a transverse averaging operator that produces canonical invariant representatives in the basic cohomology of Riemannian foliations.
• The operator facilitates isomorphisms between diffeological de Rham cohomology and Lie algebra cohomology for homogeneous spaces with non-closed subgroup actions.
• The work bridges classical averaging techniques with diffeological frameworks, extending cohomological methods to singular and non-traditional geometric settings.

Transverse Averaging Operators and Cohomology of Quotients by Non-closed Subgroups

Overview and Technical Framework

Yi Lin's paper introduces and analyzes a novel transverse averaging operator for basic differential forms on Riemannian foliations admitting isometric transverse Lie algebra actions, especially when the leaf closure space is compact. This analytic tool is then deployed to achieve new calculations and isomorphism results in the diffeological de Rham cohomology of homogeneous spaces $G/H$ where $G$ is a connected Lie group (not necessarily compact) and $H$ is a connected, possibly non-closed Lie subgroup.

The key technical innovation is a transverse averaging operator built from infinitesimal transverse data, as opposed to averaging over compact Lie group actions. This approach is applicable in the absence of global group actions and notably in settings where homogeneous spaces are defined by non-closed subgroups. The operator preserves closedness and cohomology classes, mapping closed basic forms to invariant representatives.

The work synthesizes advanced machinery from Molino’s theory, Sergiescu’s transverse integration, Poincaré duality for basic cohomology, and the structure of Riemannian foliations with isometric transverse actions.

Construction of the Transverse Averaging Operator

The classical averaging technique in equivariant geometry exploits the existence of a compact Lie group action to produce invariant differential forms representing given cohomology classes. In the foliated setting with only an isometric transverse Lie algebra action (i.e., no global group action), such averaging is unavailable.

Lin constructs the transverse averaging operator $\mathcal{A}$ as follows:

• The isometric transverse action lifts to the transverse orthonormal frame bundle $P$ over $M$.
• Molino’s machinery provides a compact “Molino manifold” $W$ upon which the simply connected Lie group $G$ with Lie algebra $\mathfrak{g}$ (the transverse structural algebra) acts by isometries. The closure $N$ of the action image in the isometry group of $G$0 is a compact, connected Lie group.
• The action of $G$1 is used to average basic forms, but crucially, this averaging is performed on the level of $G$2 (using Sergiescu’s integration theory and the duality between basic and twisted basic cohomology), and then traced back to the original foliated space.

The final operator $G$3 is defined by integrating pullbacks of forms under the $G$4-action, ensuring commutation with exterior differentiation and invariance properties required for subsequent applications.

Cohomological Applications and Main Theorems

Resolution of Basic Cohomology via Averaging: The operator $G$5 satisfies that every closed basic form $G$6 is cohomologous to its invariant average $G$7, providing canonical invariant representatives. This enables a cleaner description of the $G$8-term in the spectral sequence for equivariant basic cohomology in the Cartan model:

$G$9

This description generalizes previous results that were otherwise restricted to situations with global group actions, such as Killing foliations.

Lie Foliations of Compact Type: The operator is exploited to study Lie $H$0-foliations of compact type (where $H$1 admits a bi-invariant metric) with compact leaf closure space, yielding the isomorphism of basic cohomology to the Lie algebra cohomology:

$H$2

This result holds without the global group action, depending on the structural properties of the transverse action and Molino’s theory.

Cohomology of Homogeneous Spaces for Non-closed Subgroups:

• If $H$3 is connected, $H$4 is a connected Lie subgroup (not necessarily closed), $H$5 is of compact type, and $H$6 is compact, then

$H$7

where the left side refers to the diffeological de Rham cohomology and the right to relative Lie algebra cohomology.

• If, in addition, $H$8 is an ideal in $H$9, the assumption of compact type can be dropped:

$\mathcal{A}$0

These theorems strictly generalize the classical Chevalley--Eilenberg result; Lin's isomorphisms hold for homogeneous quotients by non-closed subgroups, under mild compactness assumptions, i.e., for any quotient with compact $\mathcal{A}$1 rather than requiring $\mathcal{A}$2 to be closed.

Rigorous Analysis and Counterexamples

The author provides explicit counterexamples demonstrating the necessity of the compactness assumption on $\mathcal{A}$3. Even for point foliations and Lie algebras not of compact type, the identified isomorphisms can fail, reinforcing the sharpness and boundaries of the main theorems.

Positive examples are also constructed in the regime of non-closed, non-dense subgroups, with $\mathcal{A}$4 noncompact and $\mathcal{A}$5 nonabelian, establishing the breadth of applicability of the results.

Diffeological Perspective and Equivariant Extensions

The computations above employ a rigorous diffeological framework for quotient spaces, given the lack of manifold structure for $\mathcal{A}$6 when $\mathcal{A}$7 is not closed. The work leverages the theorem of Hector, Macías-Virgós, and Sanmartín-Carbón linking basic cohomology of foliations to the diffeological de Rham cohomology of leaf spaces, providing precise functorial and cochain-level isomorphisms.

Further, the appendix extends these ideas to equivariant settings and establishes new results for quotients by countable groups in diffeology, which feed back into the main body to allow for generalizations to quotients $\mathcal{A}$8 where $\mathcal{A}$9 is only a dense Lie subgroup.

Implications and Future Directions

Lin’s development of a transverse averaging operator reframes how equivariant techniques can be generalized away from group actions to infinitesimal, Lie algebraic contexts. Especially in the context of foliations and non-traditional homogeneous spaces (including non-Hausdorff quotients in diffeology), it provides computable models for de Rham and basic cohomology.

This framework enables new calculations of invariants for singular spaces encountered in geometric representation theory, analysis on foliated spaces, and the study of groupoids. The operator is expected to have further implications for equivariant localization, index theory on singular quotients, and extensions to other equivariant cohomological theories in the diffeological and stack-theoretic setting.

Conclusion

This paper establishes the existence and utility of a transverse averaging operator for basic forms in Riemannian foliations, solving the long-standing obstacle of the absence of global compact group actions. Through this construction, cohomological invariants for a broad class of homogeneous spaces $P$0, including those with non-closed subgroups, are computed and shown to coincide with appropriate Lie-theoretic cohomologies under natural geometric hypotheses. This work decisively extends the scope of classical averaging and equivariant techniques within foliation theory, Lie theory, and diffeological geometry.