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 of -currents on a manifold is defined as the dual space, in the sense of distributions, of the space of -forms on . Thus there is a pairing between -currents and -forms , denoted here by
Under this duality pairing, the exterior derivative
goes over to a boundary operator
defined by
for all . This is a homological rather than cohomological construction.