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

${\displaystyle \langle T,\alpha \rangle .}$

Under this duality pairing, the exterior derivative

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

goes over to a boundary operator

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

defined by

${\displaystyle \langle \partial T,\alpha \rangle =\langle T,d\alpha \rangle }$

for all α ∈ 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.