Current (mathematics)

Homological theory

Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by ${\displaystyle [[M]]}$: $${\displaystyle [[M]](\omega )={\int }_M\omega .}$$

If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has: $${\displaystyle [[\mathrm{\partial }M]](\omega )={\int }_{\mathrm{\partial }M}\omega ={\int }_{M}d\omega =[[M]](d\omega ).}$$

This relates the exterior derivative d with the boundary operator ∂ on the homology of M.

In view of this formula we can define a boundary operator on arbitrary currents $${\displaystyle \mathrm{\partial }:{\mathcal{D}}_{m+1}\to {\mathcal{D}}_m}$$ via duality with the exterior derivative by $${\displaystyle (\mathrm{\partial }T)(\omega ):=T(d\omega )}$$ for all compactly supported m-forms ${\displaystyle \omega .}$

Certain subclasses of currents which are closed under ${\displaystyle \mathrm{\partial }}$ can be used instead of all currents to create a homology theory, which can satisfy the Eilenberg–Steenrod axioms in certain cases. A classical example is the subclass of integral currents on Lipschitz neighborhood retracts.

Topology and norms

The space of currents is naturally endowed with the weak-* topology, which will be further simply called weak convergence. A sequence ${\displaystyle T_k}$ of currents, converges to a current ${\displaystyle T}$ if $${\displaystyle T_k(\omega )\to T(\omega ),\mathrm{\forall }\omega .}$$

It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If ${\displaystyle \omega }$ is an m-form, then define its comass by $${\displaystyle \Vert \omega \Vert :=sup\{\left|⟨\omega ,\xi ⟩\right|:\xi \text{ is a unit, simple, }m\text{-vector}\}.}$$

So if ${\displaystyle \omega }$ is a simple m-form, then its mass norm is the usual L^∞-norm of its coefficient. The mass of a current ${\displaystyle T}$ is then defined as $${\displaystyle \mathbf{M}(T):=sup\{T(\omega ):\underset{x}{sup}||\omega (x)||\le 1\}.}$$

The mass of a current represents the weighted area of the generalized surface. A current such that M(T) < ∞ is representable by integration of a regular Borel measure by a version of the Riesz representation theorem. This is the starting point of homological integration.

An intermediate norm is Whitney's flat norm, defined by $${\displaystyle \mathbf{F}(T):=inf\{\mathbf{M}(T-\mathrm{\partial }A)+\mathbf{M}(A):A\in {\mathcal{E}}_{m+1}\}.}$$

Two currents are close in the mass norm if they coincide away from a small part. On the other hand, they are close in the flat norm if they coincide up to a small deformation.

Examples

Recall that $${\displaystyle {\mathrm{\Omega }}_c^0({\mathbb{R}}^n)\equiv C_c^{\mathrm{\infty }}({\mathbb{R}}^n)}$$ so that the following defines a 0-current: $${\displaystyle T(f)=f(0).}$$

In particular every signed regular measure ${\displaystyle \mu }$ is a 0-current: $${\displaystyle T(f)=\int f(x)d\mu (x).}$$

Let (x, y, z) be the coordinates in ${\displaystyle {\mathbb{R}}^3.}$ Then the following defines a 2-current (one of many): $${\displaystyle T(adx\wedge dy+bdy\wedge dz+cdx\wedge dz):={\int }_0^1{\int }_0^{1}b(x,y,0)dxdy.}$$