Abstract
We describe a subspace of de Rham currents called “differential chains,” given as an inductive limit of Banach spaces. We make use of the viewpoint introduced by Mackey and developed by Whitney in which currents are treated as chains instead of cochains. This approach gives our space and its topology an intrinsic geometric definition. Moreover, our sequence of norms gives fine control over convergence and compactness, providing viable alternatives to the weak and flat topologies on currents. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in this setting, and we are also able to define a Cartesian wedge product. We conclude with an application of this framework, generalizing the Reynolds’ Transport Theorem to nonsmooth domains.
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Notes
It is this universal property which led to the name “differential chain”.
Differential chains do not form an algebra, but they are a coalgebra and coproduct dualizes to wedge product. A preprint is in preparation.
According to his own account in [23], Whitney wanted to solve a foundational question about sphere bundles and this is why he developed Geometric Integration Theory. Steenrod [20] solved the problem first, though, and Whitney stopped working on GIT [23]. However, Whitney’s then-student Eells was interested in other applications of GIT in analysis, and for a time he thought he had a workaround [1, 22], but the lack of a continuous boundary operator halted progress.
An extension of the theory to open subsets of a Riemannian manifold M is in progress. See Sect. 13 for the main ideas.
Previously called a “pointed k-chain”. We prefer not to use the term “Dirac current” since it has various definitions in the literature, and the whole point of this work is to define calculus starting with chains, setting aside the larger space of currents.
\(\hat{\mathcal {B}}_{k}^{1}\) is isomorphic to the sharp space of polyhedral chains [22].
An equivalent definition uses simplices instead of cells. Every simplex is a cell and every cell can be subdivided into finitely many simplices.
If we had started with polyhedral chains instead of Dirac chains, convergence would be in the B 0 norm.
One can also introduce Hölder conditions into the classes of differential chains and forms as follows: For 0<β≤1, replace ∥σ j∥∥α∥=∥u j ∥⋯∥u 1∥∥α∥ in definition (2.4) with ∥u j ∥β⋯∥u 1∥∥α∥ where σ j==u j ∘…∘u 1. The resulting spaces of chains \(\hat{\mathcal {B}}_{k}^{r-1+\beta }(U)\) can permit fine tuning of the class of a chain. The inductive limit of the resulting spaces is the same \(\hat{\mathcal {B}}_{k}(U)\) as before, but we can now define the extrinsic dimension of a chain \(J \in \hat{\mathcal {B}}_{k}(U)\) as \(\dim_{E}(J) := \inf\{ k+ j-1 + \beta : J \in \hat{\mathcal {B}}_{k}^{j-1+\beta }(U) \}\). For example, \(\dim_{E}(\widetilde{\sigma }_{k}) = k\) where σ k is an affine k-cell since \(\widetilde{\sigma }_{k} \in \hat{\mathcal {B}}_{k}^{1}(U)\) (set j=0,β=1), while \(\dim_{E}(\widetilde{S}) = \ln(3)/\ln(2)\) where S is the Sierpinski triangle. The dual spaces are differential forms of class B r−1+β, i.e., the forms are of class B r−1 and the (r−1) order directional derivatives satisfy a β Hölder condition. The author’s earliest work on this theory focused more on Hölder conditions, but she has largely set this aside in recent years.
Grothendieck wrote, “Some questions arise concerning a space which is an inductive limit, which often receive negative answers, even for the inductive limits of a sequence of Banach spaces, and which often present serious difficulties.” and “We remark that in practice the difficulties which we encounter in inductive limits are the ‘converse’ of those met in projective limits (the coarsest topology for which…); here it is nearly always easy to show that the space is complete, and to determine whether its bounded subsets are weakly compact or compact…, an in particular to recognize it as either a reflexive or a Montel space.” See [5], p. 138.
Joint continuity of the bilinear operators on the inductive limit is an open question.
Whitney defined the pushforward operator F ∗ on polyhedral chains and extended it to sharp chains in [22]. He proved a change of variables formula (7.2) for Lipschitz forms. However, the important relation F ∗ ∂=∂F ∗ does not hold for sharp chains since ∂ is not defined for the sharp norm. (See Proposition 8.5 below.) The flat norm of Whitney does have a continuous boundary operator, but flat forms are highly unstable. The following example modifies an example of Whitney found on p. 270 of [22] which he used to show that components of flat forms may not be flat. But the same example shows that the flat norm has other problems. The author includes mention of these problems of the flat norm since they are not widely known, and she has seen more than one person devote years trying to develop calculus on fractals using the flat norm. Example: In \(\mathbb{R}^{2}\), let \(\omega _{t}(x,y) = \begin{cases} e_{1} + e_{2} + tu, &x+y < 0\\ 0, & x+y > 0 \end{cases} \) where t≥0 and \(u \in \mathbb{R}^{2}\) is nonzero. Then ω 0 is flat, but ω t is not flat for any t>0. In particular, setting t=2, u=−e 2, we see that ⋆ω 0 is not flat.
The choice of inner product on \(\mathbb{R}^{n}\) has no significant influence on the theory. The spaces are independent of the choice, as the norms are comparable.
Some authors call u∧v+〈u,v〉 the “geometric product”. This is found within our viewpoint of differential chains as \((E_{u} + E_{u}^{\dagger})\circ(E_{v} + E_{v}^{\dagger})(0;1) = u \wedge v + \langle u,v\rangle \).
A formal treatment of support will be given in Sect. 6.3.
The inner product we chose on \(\mathbb{R}^{n}\) induces an inner product on the exterior algebra 〈⋅,⋅〉∧ using determinant (Definition 2.2). It induces an inner product on the the symmetric algebra 〈⋅,⋅〉∘ using permanent of a matrix 〈σ,τ〉∘:=per(〈u r ,v s 〉), and thus on \(\mathcal {A}_{k}^{s}(p)\) via 〈(p;σ⊗α),(p;τ⊗β)〉⊗:=〈σ,τ〉∘〈α,β〉∧. However, this inner product 〈⋅,⋅〉⊗ on \(\mathcal {A}_{k}^{s}(p)\) does not extend to a continuous inner product on \(\mathcal {A}_{k}^{s}(U)\), although it can be useful for computations on Dirac chains of arbitrary order and dimension as long as limits are not taken.
Differential chains of class \(\mathcal {B}\) were originally called chainlets. It is only recently that the author has begun to appreciate the importance of what we now call the chainlet complex, which is a subcomplex of the differential chain complex.
Feynman wrote in his autobiography [2], “That book [Advanced Calculus, by Wood] also showed how to differentiate parameters under the integral sign—it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.”
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Acknowledgements
I am grateful to Morris Hirsch for his support of this work since its inception. His helpful remarks and historical insights have been of great benefit as the theory evolved from a few geometrically appealing, but ad hoc, and difficult constructions on polyhedral chains to its current state with its universal property [14], its algebraically concise constructions on Dirac chains, and its applications to Plateau’s problem [12]. I also am very happy to thank James Yorke who has contributed to useful discussions throughout the development of this work, while Robert Kotiuga, Alain Bossavit, Alan Weinstein, and Steven Krantz are thanked for their early interest and support. Harrison Pugh’s contributions have been significant starting with his Harvard thesis [18]. Our joint paper [14] is fundamental and necessary for results of this paper to hold in the inductive limit, and gives the entire theory a solid mathematical footing in the classical setting of topological vector spaces. Finally, I am indebted to the anonymous reviewer for providing numerous helpful comments.
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Communicated by Steven G. Krantz.
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Harrison, J. Operator Calculus of Differential Chains and Differential Forms. J Geom Anal 25, 357–420 (2015). https://doi.org/10.1007/s12220-013-9433-6
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DOI: https://doi.org/10.1007/s12220-013-9433-6
Keywords
- Topological vector space
- Inductive limit
- Differential chains
- Differential forms
- Currents
- Stokes’ theorem
- Divergence theorem
- Laplace
- Dirac
- Pushforward
- Extrusion
- Prederivative