In themathematical fields ofdifferential geometry andgeometric measure theory,homological integration orgeometric integration is a method for extending the notion of theintegral tomanifolds. Rather than functions ordifferential forms, the integral is defined overcurrents on a manifold.

The theory is "homological" because currents themselves are defined by duality with differential forms. To wit, the spaceD^k ofk-currents on a manifoldM is defined as thedual space, in the sense ofdistributions, of the space ofk-formsΩ^k onM. Thus there is a pairing betweenk-currentsT andk-formsα, denoted here by

${\displaystyle ⟨T,\alpha ⟩.}$

Under this duality pairing, theexterior derivative

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

goes over to aboundary operator

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

defined by

${\displaystyle ⟨\mathrm{\partial }T,\alpha ⟩=⟨T,d\alpha ⟩}$

for allα ∈ Ω^k. This is a homological rather thancohomological construction.

• Federer, Herbert (1969),Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, vol. 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, vol. 21, Princeton, NJ and London:Princeton University Press andOxford University Press, pp. XV+387,MR 0087148,Zbl 0083.28204.

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