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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 Dk of k-currents on a manifold M is defined as the dual space, in the sense of distributions, of the space of k-forms Ωk on M. Thus there is a pairing between k-currents T and k-forms α, denoted here by

\( \langle T,\alpha \rangle . \)

Under this duality pairing, the exterior derivative

\( d:\Omega ^{{k-1}}\to \Omega ^{k} \)

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 α ∈ Ωk. This is a homological rather than cohomological construction.
References

Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, New York: Springer-Verlag New York Inc., pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325, Zbl 0176.00801.
Whitney, H. (1957), Geometric Integration Theory, Princeton Mathematical Series, 21, Princeton, NJ and London: Princeton University Press and Oxford University Press, pp. XV+387, MR 0087148, Zbl 0083.28204.

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