Globally Subanalytic Differential Forms

• Globally subanalytic differential forms are defined on manifolds where coefficient functions are definable in ℝ₍ₐₙ₎, offering a framework that preserves tame geometric properties.
• The theory employs Whitney forms and triangulation to construct relative primitives, reducing fibrewise exactness to solvable systems of linear PDEs.
• A constructible de Rham approach remedies the failure of the naïve subanalytic complex on non-compact spaces, ensuring correct cohomological equivalence.

Globally subanalytic differential forms are differential forms on globally subanalytic manifolds whose coefficients are definable in the o-minimal structure $\mathbb{R}_{an}$, with regularity ranging from continuous forms with distributional exterior derivative to definable $C^q$-forms and constructible forms. In this setting, the interaction between tame geometry, differential operators, and simplicial or cell-theoretic decompositions produces two complementary lines of theory: a relative-primitive theory for subanalytic forms along fibres of a proper triangulable map, developed through Whitney forms on prisms, and a de Rham theory showing that the naïve globally subanalytic complex can fail on non-compact manifolds whereas a constructible de Rham complex recovers classical cohomology in full generality (Brasselet et al., 2010, Huber et al., 5 Aug 2025).

1. Ambient geometric setting

A subset $S \subset \mathbb{R}^n$ is called subanalytic if for every point $x \in \mathbb{R}^n$ there is a neighborhood $U$ of $x$ in $\mathbb{R}^n$ such that

$S \cap U = \pi(A)$

for some bounded semianalytic set $A \subset \mathbb{R}^n \times \mathbb{R}^m$ and the projection $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$. It is called globally subanalytic if in addition it is definable in the o-minimal structure $C^q$0; equivalently, its image in the projective chart is subanalytic. In the later de Rham treatment, the same notion is expressed by saying that a subset $C^q$1 is globally subanalytic if it is the image of a bounded semianalytic set under a proper projection in some higher $C^q$2, equivalently definable in $C^q$3 (Brasselet et al., 2010, Huber et al., 5 Aug 2025).

A real-analytic manifold $C^q$4 is called subanalytic, or globally subanalytic, if it admits an atlas by charts whose domains and images are respectively open in $C^q$5 and globally subanalytic in some $C^q$6, with real-analytic subanalytic transition maps. More generally, the de Rham framework allows a globally subanalytic $C^q$7-manifold to be defined by a finite covering $C^q$8 together with homeomorphisms $C^q$9 onto open definable sets, such that all transition maps $S \subset \mathbb{R}^n$0 are $S \subset \mathbb{R}^n$1 and definable. Two atlases are equivalent if their union is again such an atlas (Huber et al., 5 Aug 2025).

This geometric setting is distinguished by finiteness and tame-topological properties. A plausible implication is that constructions familiar from smooth and analytic geometry can be reformulated so that they remain compatible with definability, triangulation, and parameter dependence. In the cited works, this compatibility is exploited in two different but related ways: first, to solve fibrewise exactness problems by triangulating a proper map and reducing to explicit prismwise equations; second, to define de Rham complexes whose coefficients remain within tame classes stable under the operations needed for cohomology (Brasselet et al., 2010, Huber et al., 5 Aug 2025).

2. Classes of globally subanalytic differential forms

On a real-analytic manifold $S \subset \mathbb{R}^n$2, an $S \subset \mathbb{R}^n$3-form $S \subset \mathbb{R}^n$4 is called subanalytic continuous if in every chart its coordinate functions are continuous subanalytic functions and its distributional exterior derivative $S \subset \mathbb{R}^n$5 admits a bounded subanalytic representative. Equivalently, by Proposition 3.4 of Brasselet–Teissier, $S \subset \mathbb{R}^n$6 is $S \subset \mathbb{R}^n$7-regular in Whitney’s sense, its coefficients are bounded subanalytic, and its derived form in Whitney’s sense is subanalytic; equivalently again, the distributional $S \subset \mathbb{R}^n$8 has a bounded subanalytic representative. The form is globally subanalytic if all these local representatives are globally subanalytic functions on $S \subset \mathbb{R}^n$9 (Brasselet et al., 2010).

In the $x \in \mathbb{R}^n$0-manifold framework, one has the cotangent bundle $x \in \mathbb{R}^n$1 of class $x \in \mathbb{R}^n$2 and its exterior powers $x \in \mathbb{R}^n$3. A $x \in \mathbb{R}^n$4-section of $x \in \mathbb{R}^n$5, with $x \in \mathbb{R}^n$6, is called a definable $x \in \mathbb{R}^n$7-form of degree $x \in \mathbb{R}^n$8, and the corresponding space is denoted

$x \in \mathbb{R}^n$9

If in addition the coefficient functions in every chart are constructible, one writes $U$0 (Huber et al., 5 Aug 2025).

A constructible function on a globally subanalytic set $U$1 is, by definition, a finite $U$2-linear combination of finite products of restricted analytic functions and logs of positive such. Equivalently, it is definable in $U$3 and closed under parametric integration. A $U$4-form is constructible of class $U$5 if in every chart its coefficient functions are constructible $U$6-functions (Huber et al., 5 Aug 2025).

These notions form a hierarchy rather than a single category. Continuous subanalytic forms in the sense of relative primitives are designed to accommodate distributional $U$7 and piecewise-linear simplicial models. Definable $U$8-forms provide the natural “rough” globally subanalytic de Rham complex. Constructible forms enlarge the coefficient class just enough to admit logarithmic primitives such as $U$9, which become decisive in the non-compact case. This suggests that the globally subanalytic category, by itself, is too small for a full de Rham theorem on arbitrary definable manifolds, while still being sufficiently rigid for relative and fibrewise statements.

3. Relative primitives and fibrewise exactness

A central existence theorem concerns a proper subanalytic real-analytic map $x$0 that is triangulable; an explicit example is the case where $x$1 is compact. Let $x$2 be a continuous subanalytic $x$3-form on $x$4 such that for each $x$5 at which the fibre $x$6 is a nonsingular manifold, the pull-back $x$7 is exact. Then there exists a continuous subanalytic $x$8-form $x$9 on $\mathbb{R}^n$0 such that

$\mathbb{R}^n$1

holds on $\mathbb{R}^n$2 in the sense of distributions. Brasselet and Teissier formulate this as Theorem 6.1 and Corollary 6.3 (Brasselet et al., 2010).

The geometric meaning of the relation

$\mathbb{R}^n$3

is that wedging with $\mathbb{R}^n$4 kills precisely the horizontal component of $\mathbb{R}^n$5, so that on each fibre $\mathbb{R}^n$6 one has

$\mathbb{R}^n$7

Thus $\mathbb{R}^n$8 is a relative primitive of $\mathbb{R}^n$9 along the fibres of $S \cap U = \pi(A)$0 (Brasselet et al., 2010).

The theorem does not assert a global primitive $S \cap U = \pi(A)$1 on $S \cap U = \pi(A)$2. Instead, it produces a primitive modulo horizontal terms determined by the base map $S \cap U = \pi(A)$3. This distinction is essential: the obstruction being addressed is fibrewise exactness, not absolute exactness on the total space. A common misunderstanding is to read the theorem as a Poincaré lemma in the globally subanalytic category; the statement is narrower and more geometric, because it is organized around a proper triangulable map and the behaviour of forms on its nonsingular fibres (Brasselet et al., 2010).

A basic example is the projection case. If $S \cap U = \pi(A)$4 is a compact globally subanalytic manifold and $S \cap U = \pi(A)$5 is the projection onto $S \cap U = \pi(A)$6, then any subanalytic form $S \cap U = \pi(A)$7 on $S \cap U = \pi(A)$8 whose restriction to each fibre is exact admits a subanalytic primitive $S \cap U = \pi(A)$9 on $A \subset \mathbb{R}^n \times \mathbb{R}^m$0 satisfying $A \subset \mathbb{R}^n \times \mathbb{R}^m$1. The concrete procedure is to triangulate $A \subset \mathbb{R}^n \times \mathbb{R}^m$2 so that fibres become products of simplices, write $A \subset \mathbb{R}^n \times \mathbb{R}^m$3 in the associated Whitney basis, and solve the resulting simple PDE on each prism (Brasselet et al., 2010).

4. Whitney forms, prismal sheaves, and the reduction to PDEs

The proof of the relative-primitive theorem proceeds by subanalytic triangulation adapted to the map $A \subset \mathbb{R}^n \times \mathbb{R}^m$4. By Shiota’s theorem, any proper subanalytic map from a compact analytic manifold to $A \subset \mathbb{R}^n \times \mathbb{R}^m$5 is subanalytically triangulable. One chooses a globally subanalytic homeomorphism

$A \subset \mathbb{R}^n \times \mathbb{R}^m$6

which is analytic on each simplex of a linear simplicial decomposition, together with simplicial decompositions of $A \subset \mathbb{R}^n \times \mathbb{R}^m$7 and of $A \subset \mathbb{R}^n \times \mathbb{R}^m$8, so that the projection after precomposing by $A \subset \mathbb{R}^n \times \mathbb{R}^m$9 becomes simplicial and $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$0 is carried to a subcomplex $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$1. Writing $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$2 for the induced simplicial map, one replaces the study of $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$3 on $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$4 by $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$5 on $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$6 and works over the simplicial map $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$7 (Brasselet et al., 2010).

From the simplicial map $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$8 one builds two prismal sheaves on the base $$\pi:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$9. The first, $C^q$00, has fibre over a simplex $C^q$01 equal to the simplicial preimage $C^q$02. The second, $C^q$03, has fibre over $C^q$04 equal to a prism decomposition of $C^q$05 into products of simplices, trivialized over each closed simplex of $C^q$06. On each prism

$C^q$07

of $C^q$08 one defines the relative Whitney form

$C^q$09

where $C^q$10 denotes the usual Whitney form of the simplex. These relative Whitney forms form a basis for the vertical cohomology in each fibre, specifically in Lemma 4.5 and Lemma 4.6 (Brasselet et al., 2010).

Expressing $C^q$11, modulo vertical exact forms, as a subanalytic linear combination of these relative Whitney forms, one seeks $C^q$12 of the same type. The condition

$C^q$13

on each prism becomes a finite system of linear first-order PDEs for the unknown subanalytic coefficient functions $C^q$14 on each maximal prism $C^q$15 of dimension $C^q$16. Concretely, for each face $C^q$17 of $C^q$18 with the same base image and of relative degree $C^q$19, one obtains an equation of the form

$C^q$20

where the $C^q$21 are barycentric coordinates in the vertical factors, the $C^q$22 are known affine functions, and $C^q$23 is a known subanalytic function; the paper identifies this as equation (6.12) (Brasselet et al., 2010).

The significance of this reduction is methodological. The combinatorics of Whitney forms converts a fibrewise differential problem into a linear PDE system on simplices and prisms, with coefficients governed by barycentric geometry. The global subanalytic structure enters twice: first through triangulability and prismal trivialization, and then through the requirement that the coefficient functions solving the PDE remain subanalytic.

5. The model PDE and subanalytic integral formulas

On the standard $C^q$24-simplex

$C^q$25

the prototype equation is

$C^q$26

with $C^q$27 subanalytic continuous and analytic in the interior. Proposition 6.2 shows that this equation has the unique continuous subanalytic solution

$C^q$28

If $C^q$29 is arc-analytic, then $C^q$30 is also arc-analytic (Brasselet et al., 2010).

The crucial point is not only the formal integral representation, but the preservation of subanalyticity. By rectilinearization, or toric resolution, of subanalytic functions, the parameter-dependent integral

$C^q$31

is again subanalytic and continuous. Analyticity in the interior follows from term-by-term convergence of the power series of $C^q$32 under the integral (Brasselet et al., 2010).

This step closes the proof of the relative-primitive theorem: once the prismwise coefficients satisfy the model first-order equations, the integral formula yields coefficient functions that remain in the subanalytic category, thereby producing a globally defined continuous subanalytic primitive $C^q$33. The summary given in the source emphasizes the interplay among global subanalytic geometry, Whitney-form combinatorics, and a linear PDE whose solution has a subanalytic integral representation; it also states that the final primitive is subanalytic and continuous, even Hölder (Brasselet et al., 2010).

A plausible implication is that, within tame geometry, many parameter-dependent primitive constructions depend less on elliptic or microlocal machinery than on the stability of definable classes under triangulation, barycentric formulas, and integration along controlled rays.

6. De Rham complexes, failure of the naïve theory, and the constructible remedy

For a globally subanalytic $C^q$34-manifold $C^q$35, one may define the rough subanalytic de Rham complex by taking

$C^q$36

for $C^q$37, with $C^q$38 the usual exterior derivative, which lowers differentiability by one. This produces a complex

$C^q$39

with cohomology

$C^q$40

However, this complex does not in general compute the correct cohomology for non-compact $C^q$41 (Huber et al., 5 Aug 2025).

The basic counterexample is $C^q$42. The closed $C^q$43-form

$C^q$44

is not exact in the subanalytic category, because any primitive would be $C^q$45, which is not globally subanalytic. Hence

$C^q$46

whereas singular cohomology satisfies $C^q$47 (Huber et al., 5 Aug 2025).

The constructible theory modifies the coefficient class rather than the cohomological formalism. A constructible $C^q$48-form has coefficient functions that are constructible, and one defines $C^q$49 analogously. For $C^q$50, one requires $C^q$51; for $C^q$52, one uses a dense open $C^q$53-zone and extends Cartan differential continuously. This yields a subcomplex

$C^q$54

with cohomology denoted $C^q$55 (Huber et al., 5 Aug 2025).

The principal theorem states that for any possibly non-compact globally subanalytic $C^q$56-manifold $C^q$57 with $C^q$58, and for every $C^q$59, the pairing

$C^q$60

is perfect. In particular, there is a canonical isomorphism

$C^q$61

and if $C^q$62 is real-analytic, then $C^q$63 by the classical analytic de Rham theorem (Huber et al., 5 Aug 2025).

The proof uses standard de Rham-theoretic ingredients in a definable form: functoriality of pullback for constructible $C^q$64-forms, Stokes’ theorem, homotopy invariance via the operator

$C^q$65

the Poincaré lemma on cells obtained from a finite cell decomposition into ribbons, and a Mayer–Vietoris induction using definable $C^q$66-partitions of unity (Huber et al., 5 Aug 2025).

An important point of emphasis is regularity. The constructible de Rham theorem already holds in the minimal regularity $C^q$67: all constructions are arranged so that no higher smoothness is needed to build a valid de Rham theory in the constructible category. This marks a sharp contrast with the relative-primitive theory, which is stated for real-analytic manifolds and proper subanalytic real-analytic maps (Brasselet et al., 2010, Huber et al., 5 Aug 2025).

7. Interrelations, examples, and scope

The two strands of the subject address different questions. The relative-primitive theorem concerns a form on a total space together with a proper triangulable map $C^q$68, and asks whether fibrewise exactness can be integrated into a global subanalytic form $C^q$69 satisfying $C^q$70. The constructible de Rham theorem concerns the cohomology of the manifold itself and asks which coefficient class yields a de Rham complex equivalent to singular cohomology (Brasselet et al., 2010, Huber et al., 5 Aug 2025).

They are nevertheless linked by common techniques and constraints. In the relative theory, the integration of coefficient functions over rays in simplices preserves subanalyticity and yields explicit primitives. In the constructible theory, closure under parametric integration is built into the coefficient class itself. This suggests that parametric integration is one of the decisive operations controlling whether a tame coefficient category is cohomologically adequate.

The consequences stated for subanalytic chains and fibres reinforce this connection. As a corollary of the relative-primitive theory, one obtains well-defined subanalytic functions

$C^q$71

on the parameter $C^q$72, provided the degree matches the fibre dimension; more generally, for a subanalytic chain $C^q$73, one can define $C^q$74 as a subanalytic function of additional parameters. The source attributes these developments to a context including Lion–Rolin (Brasselet et al., 2010).

A common misconception is that “globally subanalytic differential forms” form a single de Rham category with the same formal properties as smooth forms on manifolds. The available results show a more differentiated picture. Continuous and definable globally subanalytic forms are robust enough for triangulation-based primitive constructions and for many local differential operations, but the naïve globally subanalytic de Rham complex fails on non-compact spaces. Constructible forms repair this failure by enlarging the coefficient class in a controlled way while preserving definability and admitting a perfect pairing with singular homology (Huber et al., 5 Aug 2025).

Within this framework, globally subanalytic differential forms occupy an intermediate position between real-analytic geometry and o-minimal topology. The subject combines triangulability, Whitney-form combinatorics, distributional or low-regularity differential calculus, cell decomposition, and parameter-stable integration. The resulting theory is therefore not a single theorem but a structured collection of formalisms adapted to two distinct objectives: relative primitives in the subanalytic category and cohomological equivalence in the constructible one.