- Source: 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 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
⟨
T
,
α
⟩
.
{\displaystyle \langle T,\alpha \rangle .}
Under this duality pairing, the exterior derivative
d
:
Ω
k
−
1
→
Ω
k
{\displaystyle d:\Omega ^{k-1}\to \Omega ^{k}}
goes over to a boundary operator
∂
:
D
k
→
D
k
−
1
{\displaystyle \partial :D^{k}\to D^{k-1}}
defined by
⟨
∂
T
,
α
⟩
=
⟨
T
,
d
α
⟩
{\displaystyle \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, 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.
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