Homological integration

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 $k$ -currents on a manifold $M$ is defined as the dual space, in the sense of distributions, of the space of $k$ -forms $Ω k$ 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 $α \in Ω k$ . This is a homological rather than cohomological construction.

References

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 and Oxford University Press, pp. XV+387, MR 0087148, Zbl 0083.28204.