Homological integration

This article is about an extension of the theory of the Lebesgue integral to manifolds. For numerical method, see 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 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 ∈ 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.