Homological integration explained

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

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 . This is a homological rather than cohomological construction.

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