Homological integration

In the mathematical fields of differential geometry and geometric measure theory, homological integration or geometric integration is a method for extending the notion of the integral to manifolds. Rather than functions or differential forms, the integral is defined over currents on a manifold.

The theory is "homological" because currents themselves are defined by duality with differential forms. To wit, the space $D^{k}$ of $k$-currents on a manifold $M$ is defined as the dual space, in the sense of distributions, of the space of $k$-forms $Ω^{k}$ on $M$. Thus there is a pairing between $k$-currents $T$ and $k$-forms $α$, denoted here by
 * $$\langle T, \alpha\rangle.$$

Under this duality pairing, the exterior derivative
 * $$d : \Omega^{k-1} \to \Omega^k$$

goes over to a boundary operator
 * $$\partial : D^k \to D^{k-1} $$

defined by
 * $$\langle\partial T,\alpha\rangle = \langle T, d\alpha\rangle$$

for all $α ∈ Ω^{k}$. This is a homological rather than cohomological construction.