6.1 Flat chains and cochains

Flat chains and cochains are powerful tools in geometric measure theory. They extend the concept of submanifolds to include more complex objects like fractals. These mathematical structures form normed abelian groups, allowing for rigorous analysis of geometric properties.

The flat norm and its dual provide a framework for studying convergence and compactness. This enables us to tackle challenging problems in geometry and analysis, bridging the gap between smooth and discrete structures in mathematics.

Flat chains and cochains

Definition and notation

• Flat chains generalize oriented submanifolds with integer multiplicities, allowing for more general objects like fractals and non-oriented sets
• Flat chains form a normed abelian group, denoted by $\mathcal{F}_k(U)$ for $k$-dimensional flat chains in an open set $U \subset \mathbb{R}^n$
• The flat norm of a flat chain $T \in \mathcal{F}_k(U)$ is defined as $\mathbf{F}(T) = \inf\{\mathbf{M}(P) + \mathbf{M}(Q) : T = P + \partial Q, P \in \mathcal{F}_k(U), Q \in \mathcal{F}_{k+1}(U)\}$, where $\mathbf{M}$ denotes the mass norm
• Flat cochains are the dual objects to flat chains, belonging to the dual space $\mathcal{F}^k(U) = (\mathcal{F}_k(U))^*$
• The pairing between a flat chain $T \in \mathcal{F}_k(U)$ and a flat cochain $\omega \in \mathcal{F}^k(U)$ is denoted by $\langle T, \omega \rangle$

Mass and norms

• The mass of a flat chain $T \in \mathcal{F}_k(U)$ is defined as $\mathbf{M}(T) = \sup\{T(\omega) : \omega \in \mathcal{D}^k(U), \|\omega\|_\infty \leq 1\}$, where $\mathcal{D}^k(U)$ denotes the space of smooth differential $k$-forms with compact support in $U$
• The flat norm on chains induces a dual norm on cochains, given by $\|\omega\|_{\mathcal{F}^k(U)} = \sup\{|\langle T, \omega \rangle| : T \in \mathcal{F}_k(U), \mathbf{F}(T) \leq 1\}$
• The flat norm and dual norm provide a natural framework for studying convergence and compactness properties in geometric measure theory

Properties of flat chains and cochains

Boundary and coboundary operators

• The boundary operator $\partial: \mathcal{F}_k(U) \to \mathcal{F}_{k-1}(U)$ satisfies $\partial \circ \partial = 0$ and is a continuous linear map with respect to the flat norm
• The coboundary operator $\delta: \mathcal{F}^k(U) \to \mathcal{F}^{k+1}(U)$ is the dual operator to $\partial$, defined by $\langle \partial T, \omega \rangle = \langle T, \delta \omega \rangle$ for all $T \in \mathcal{F}_k(U)$ and $\omega \in \mathcal{F}^{k-1}(U)$
• Flat chains and cochains form a chain complex $(\mathcal{F}_*(U), \partial)$ and a cochain complex $(\mathcal{F}^*(U), \delta)$, respectively

Integration and differential forms

• The integration of a differential form over a submanifold defines a flat cochain, providing a link between geometric measure theory and differential forms
• Flat cochains can be used to study the geometric properties of chains through their action on cochains, and vice versa

Examples of flat chains and cochains

Low-dimensional examples

• In $\mathbb{R}^1$, a flat 0-chain is a finite sum of Dirac measures with integer coefficients (point masses), while a flat 1-chain is a rectifiable curve with integer multiplicities (weighted curves)
• In $\mathbb{R}^2$, a flat 1-chain can represent a network of curves or a graph with integer edge weights (weighted networks), while a flat 2-chain can represent a piecewise smooth surface with integer multiplicities (weighted surfaces)

Cochains and functions

• A flat 0-cochain in $\mathbb{R}^n$ is a locally integrable function (regular functions)
• A flat $n$-cochain in $\mathbb{R}^n$ is a signed Radon measure (generalized functions or distributions)

Flat chains vs dual spaces

Isometric isomorphism

• The dual space of $\mathcal{F}_k(U)$ is isometrically isomorphic to the completion of $\mathcal{D}^k(U)$ with respect to the dual norm induced by the flat norm on chains
• This isomorphism allows for the study of flat chains through their action on smooth differential forms, and vice versa

Hahn-Banach theorem and norm-attaining cochains

• The Hahn-Banach theorem implies that for every flat cochain $\omega \in \mathcal{F}^k(U)$, there exists a flat chain $T \in \mathcal{F}_k(U)$ such that $\langle T, \omega \rangle = \|\omega\|_{\mathcal{F}^k(U)}$ and $\mathbf{F}(T) = 1$
• This result demonstrates the duality between flat chains and cochains, showing that every cochain can be "realized" by a flat chain with unit flat norm