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Geometric Poincar\'e Lemma

A geometric version of the Poincar\'e Lemma is established for the topological vector space of differential chains. In particular, every differential k-cycle with compact support in a contractible open subset U of a smooth n-manifold M is the boundary of a differential (k+1) -chain with compact support in U. Applications include generalizations of the Intermediate Value Theorem and Rolle's Theorem.


Introduction
The Poincaré Lemma for the de Rham complex Ω(U ) = ⊕ n k=0 Ω k (U ) of smooth differential forms defined on contractible open subsets U of R n is a foundational result with applications from topology to physics: Every closed differential form ω ∈ Ω k (U ) is exact. That is, if dω = 0 where ω ∈ Ω k (U ), then there exists η ∈ Ω k−1 (U ) such that ω = dη. Such forms η are found through a homotopy operator A acting on forms where η = Aω. That such an operator A exists is a remarkable and powerful feature of the de Rham complex.
A homotopy operator K on the chain complex of polyhedral k-chains is easy to construct using the classical cone construction from topology, and works quite well for the category of polyhedral chains. One way to generalize Poincaré's Lemma is to introduce a topology on each vector space of polyhedral k-chains with the hope of extending K to a continuous homotopy operator on the chain complex of the completed spaces. For a coherent theory with broad application to domains of integration going far beyond polyhedral chains, we need the dual space to be an identifiable space of differential k-forms, and the dual operator Aω := ωK, which is necessarily continuous since K is continuous, to be a computable homotopy operator.
Whitney's Banach space of "sharp chains" [?] has no continuous boundary operator, and thus there is no meaningful cone operator in the sharp space. The cone operator K on polyhedral chains is not generally continuous in the Banach space of "flat chains", although it is continuous in the subspace of flat chains with finite mass, and this has been useful as a tool for solving a special case of Plateau's problem for integral currents ( [?], see also [?]). Our efforts to solve the Plateau problem in full generality [?] led to the results in this paper.
Our main result is a geometric Poincaré Lemma for a certain differential chain complex (to be defined below) of topological vector spacesB k (U 1 ) of "differential k-chains" in an open set U 1 that As a corollary, we obtain the following dual result. Let B k (U ) be the Frechét space of differential forms defined on U , each with uniforms bounds on each of its directional derivatives. Then B k (U ) is the topological dual ofB k (U ) (see [?] Theorem 2.12.8).
There are three primary reasons for calling our result a "geometric Poincaré Lemma". First of all, the topology ofB k (U ) is defined constructively, and is not simply the abstract dual of the space of currents B k (U ) . Similarly, the cone operator K is defined geometrically. Finally, Dirac chains are dense inB k (U ), yielding a discrete and computable version of the geometric Poincaré's lemma and its cone operator K. The operator K restricts to the classical cone operator on polyhedral chains, and thus our theorem is a generalization of the classical version of Poincaré's Lemma for chains.
Applications presented in this paper include a broad generalization of the Intermediate Value Theorem to arbitrary dimension and codimension (see Theorem 4.1.1 and Figure 1), a new approach to homology theory with potential extensions to non-manifolds [?]. H. Pugh uses results of this paper to find generalizations of the Cauchy integral theorems [?], and, in a sequel [?], the author uses the geometric Poincaré Lemma to provide a general solution to Plateau's problem.
The author is indebted to M.W. Hirsch and H. Pugh for helpful comments.

Preliminaries
This work relies on new methods of calculus presented in [?]. In this preliminary section we recount the definition of the topological vector spaceB(U ) and its operators from [?] which we use below.
2.1. The Mackey topology on Dirac chains. Let A k (U ) be the vector space of Dirac k-chains, i.e., formal sums (p i ; α i ) where p i ∈ U and α i ∈ Λ k (R n ). We call (p; α) a simple k-element if α is a simple k-vector. Otherwise, (p; α) is called a k-element. Since (A k (U ), B k (U )) is a dual pair, the Mackey topology τ k is uniquely determined on A k (U ). (The Mackey topology τ k is the finest topology µ k on A k (U ) such that (A k (U ), µ k ) = B k (U ).) This process may be mimicked for any subspace of forms, e.g., D k (U ), the space of smooth functions with compact support in U , but B k (U ) is especially nice to work with because of its useful algebra of continuous operators, and algebraic features of its topology. The complexB k (U ), which can also be defined for open subsets of Riemannian manifolds, is intrinsic in the following sense recently announced in [?]: Theorem. The topology τ k (U ) is the finest topology in the collection {µ} of locally convex Hausdorff topologies on Dirac chains A k (U ) satisfying three axioms: (a) The topological vector space (A k (U ), µ) is bornological; (p; α/t) is well-defined and uniformly bounded for all unit vectors v ∈ R n .
The first two properties are essential to analysis in this space, the third suffices for the Lie derivative of differential forms to be a bounded operator. It is shown in [?] that the Mackey topology τ k coincides with the constructive definition of τ k of [?], which we next recall.
2.2. Constructive definition of the B r norms.
2.2.1. Mass norm. An inner product ·, · on R n determines the mass norm on Λ k (U ) as follows: Let The mass of a simple k-vector α = v 1 ∧· · ·∧v k is defined by Mass is a norm on the subspace of Dirac k-chains supported in p, since that subspace is isomorphic to the exterior algebra Λ(R n ) = ⊕ n k=0 Λ k (R n ) for which mass is a norm (see [?], p 38-39). The mass of a Dirac k- If a different inner product is chosen, the resulting masses of Dirac chains are topologically equivalent.
It is straightforward to show that · B 0 is a norm on A k (U ).

2.2.2.
Difference chains and the B r norm on chains. Given a k-element (p; α) with p ∈ U and u ∈ R n , let T u (p; α) := (p + u; α) be translation through u, and ∆ u (p; α) := (T u − I)(p; α). Let S j = S j (R n ) be the j-th symmetric power of the symmetric algebra S(R n ). Denote the symmetric product in the symmetric algebra S(R n ) by ). Let σ := u 1 · · · u j and |∆ σ j (p; α)| B j := σ α . Let ∆ σ 0 (p; α) := (p; α), to keep the notation consistent. We say ∆ σ j (p; α) For simplicity, we often write It is easy to see that the B r norms on Dirac chains are decreasing as r increases.
It is shown in [?] (Theorem 2.6.1) that · B r is a norm on the free space of Dirac k-chains A k (U ) be the inductive limit as r → ∞, endowed with the inductive limit topology. It is shown in [?] thatB k (U ) ∼ = (A k (U ), τ k ). Therefore, the inductive limit topology coincides with the Mackey topology τ k .
the Banach space of Lipschitz k-forms, and for each r > 1, B r k (U ) be the Banach space of differential k-forms, each with bounds on each of the s-th order directional derivatives for 0 ≤ s ≤ r − 1 and the r-th derivatives satisfy a Lipschitz condition. Denote the resulting norm by · B r . Elements of B r k (U ) are called differential k-forms of class B r in U . The natural inclusionsB r k (U 1 ) →B r k (U 2 ) are continuous for all open U ) be the projective limit as r → ∞, endowed with the Frechét topology. If U 1 ⊂ U 2 , the natural restrictions B r k (U 2 ) → B r k (U 1 ) are continuous, but they are not surjective unless U 1 is a regular open subset, that is, U 1 equals the interior of its closure. It is shown in [?] Theorem 2.8.2 that (B r k (U )) ∼ = B r k (U ), and the integral pairing - If J is nonzero then its support supp(J ) is a uniquely determined nonempty set (see [?] Theorem Is S :B(U 2 ) →B(U 1 ) closed? The constructive definition of the Mackey topology τ k is often useful for answering such questions as seen in the following Lemma.
Boundary ∂ = ∂ k coincides with the classical boundary operator ∂ k on polyhedral k-chains, which are dense inB k (U ), although a direct definition using "higher order Dirac chains" is provided in [?] §4. We say a differential k-chain γ ∈B k (U ) is a differential k-cycle if ∂γ = 0.
Let M r (U, R m ) be the vector space of differentiable maps F : U → R m so that the directional derivatives L ej F i of its coordinate functions F i are of class B r−1 , for r ≥ 1. Define the seminorm We write |F | D r = |F | D r,U when U is understood. It is not hard to show that |·| D r is a seminorm, but Theorem 2.4.2. If F ∈ M r (U 1 , U 2 ), then for all A ∈ A 0 (U 1 ) and all r ≥ 0. and thus determines a continuous linear map For a proof see [?] Theorems 7.5.4 and 7.5.5.
Theorem 2.4.3. Any affine k-cell τ in U is represented by a unique differential k-chain τ ∈B k (U ) such that τ ω = τ ω for all ω ∈ B 1 k (U ).
bilinear and continuous for all open sets U 1 ⊂ R n , U 2 ⊂ R m and satisfies See [?] Theorem 10.1.3.

Poincaré Lemma
is a bigraded chain complex. Letting s → ∞, the inductive limits form a chain complex since ∂ k is well-defined and continuous on the inductive limits.
A collection of maps S k : Definition 3.1.1. For U ⊂ R n open, the k-th "differential homology" group of the chain complex 2 By the universal property of tensor product, × factors through a continuous linear map cross product× : where the inductive limit is taken as → 0.
If F k :B r k (U 1 ) →B r k (U 2 ) is a graded map of chain complexes, then F k induces a well-defined map Two graded maps of complexes F k , G k :B r k (U 1 ) →B r k (U 2 ) are chain homotopic through a family of maps It is a standard result that if two maps of complexes F k , G k are chain homotopic, thenĤ k (F k ) =Ĥ k (G k ).
We may now state our main result. where U 1 is B r contractible in U 2 . Then for J ∈B r k (U 1 ) with ∂J = 0, there exists C ∈B r k+1 (U 2 ) with ∂C = J for all 1 ≤ k ≤ n and 1 ≤ r ≤ ∞.
In order to prove this we first establish Corollary 3.1.6 which says that pushforwards through homotopic maps are homotopic as maps of complexes.
Theorem 3.1.4. If F ∈ M r ([0, 1] × U 1 → U 2 ), then K = F * L extends to a continuous linear map Theorem 3.1.5. If f 0 , f 1 : U 1 ⊂ R n → U 2 ⊂ R m are B r homotopic, the maps of chain complexes f 1 * , f 0 * :B r k (U 1 ) →B r k (U 2 ) are chain homotopic through the family of maps {K k }. That is, Proof. By Theorems 2.4.2 and 2.5.1 Proof of the Geometric Poincaré Lemma:Ĥ k (c * ) = 0 and thusĤ k (I * ) = 0. But this implieŝ H k (U 1 ) = 0. That is, im ∂ = ker ∂ which is what we wanted to prove.
The cone over a differential cycle is unique up to addition of a differential chain boundary since J = ∂(KJ + ∂C). The chain ∂C is a gauge choice. In codimension one, L = 0.

Discrete Poincaré Lemma
Discrete versions of the Poincaré Lemmas are readily available since Dirac chains are dense in the space of differential chains. If J ∈B k (U ), we can approximate J with A = (p i ; α i ) and apply the operator K to each k-element (p i ; α i ). Since K is linear and continuous, KJ is approximated by K(p i ; α i ). The differential complex A j k (U ) of Dirac chains of arbitrary order and dimension in U (see [?] §3.4) is the discrete analogue ofB k (U ). Since the other operators we use, e.g., boundary, are also linear and continuous, we obtain discrete versions of all of the results of this paper. If we fix a finite set of "base points" {p i }, the space of Dirac chains becomes finite dimensional and the operators can be represented as matrices.