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

Under this duality pairing, the exterior derivative

goes over to a boundary operator

defined by

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