Homological integration
# 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.

