Homological integration

Template:Short description 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 Template:Mvar-currents on a manifold Template:Mvar is defined as the dual space, in the sense of distributions, of the space of Template:Mvar-forms Ω^k on Template:Mvar. Thus there is a pairing between Template:Mvar-currents Template:Mvar and Template:Mvar-forms Template:Mvar, denoted here by

$⟨T,\alpha ⟩.$

Under this duality pairing, the exterior derivative

$d:{\mathrm{\Omega }}^{k-1}\to {\mathrm{\Omega }}^k$

goes over to a boundary operator

$\partial :D^k\to D^{k-1}$

defined by

$⟨\partial T,\alpha ⟩=⟨T,d\alpha ⟩$

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

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