homological integration

{{Short description|Mathematics concept}}

{{about|an extension of the theory of the Lebesgue integral to manifolds|numerical method|geometric integrator}}

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 {{math|D^k}} of {{mvar|k}}-currents on a manifold {{mvar|M}} is defined as the dual space, in the sense of distributions, of the space of {{mvar|k}}-forms {{math|Ω^k}} on {{mvar|M}}. Thus there is a pairing between {{mvar|k}}-currents {{mvar|T}} and {{mvar|k}}-forms {{mvar|α}}, denoted here by

:$\backslash langle\; T,\; \backslash alpha\backslash rangle.$

Under this duality pairing, the exterior derivative

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

goes over to a boundary operator

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

defined by

:$\backslash langle\backslash partial\; T,\backslash alpha\backslash rangle\; =\; \backslash langle\; T,\; d\backslash alpha\backslash rangle$

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