Current (mathematics)

(Redirected from Homological current)

In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of M.

Definition

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Let   denote the space of smooth m-forms with compact support on a smooth manifold   A current is a linear functional on   which is continuous in the sense of distributions. Thus a linear functional   is an m-dimensional current if it is continuous in the following sense: If a sequence   of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when   tends to infinity, then   tends to 0.

The space   of m-dimensional currents on   is a real vector space with operations defined by  

Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current   as the complement of the biggest open set   such that   whenever  

The linear subspace of   consisting of currents with support (in the sense above) that is a compact subset of   is denoted  

Homological theory

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Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by  :  

If the boundaryM of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has:  

This relates the exterior derivative d with the boundary operator ∂ on the homology of M.

In view of this formula we can define a boundary operator on arbitrary currents   via duality with the exterior derivative by   for all compactly supported m-forms  

Certain subclasses of currents which are closed under   can be used instead of all currents to create a homology theory, which can satisfy the Eilenberg–Steenrod axioms in certain cases. A classical example is the subclass of integral currents on Lipschitz neighborhood retracts.

Topology and norms

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The space of currents is naturally endowed with the weak-* topology, which will be further simply called weak convergence. A sequence   of currents, converges to a current   if  

It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If   is an m-form, then define its comass by  

So if   is a simple m-form, then its mass norm is the usual L-norm of its coefficient. The mass of a current   is then defined as  

The mass of a current represents the weighted area of the generalized surface. A current such that M(T) < ∞ is representable by integration of a regular Borel measure by a version of the Riesz representation theorem. This is the starting point of homological integration.

An intermediate norm is Whitney's flat norm, defined by  

Two currents are close in the mass norm if they coincide away from a small part. On the other hand, they are close in the flat norm if they coincide up to a small deformation.

Examples

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Recall that   so that the following defines a 0-current:  

In particular every signed regular measure   is a 0-current:  

Let (x, y, z) be the coordinates in   Then the following defines a 2-current (one of many):  

See also

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Notes

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References

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  • de Rham, Georges (1984). Differentiable manifolds. Forms, currents, harmonic forms. Grundlehren der mathematischen Wissenschaften. Vol. 266. Translated by Smith, F. R. With an introduction by S. S. Chern. (Translation of 1955 French original ed.). Berlin: Springer-Verlag. doi:10.1007/978-3-642-61752-2. ISBN 3-540-13463-8. MR 0760450. Zbl 0534.58003.
  • Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. Berlin–Heidelberg–New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. ISBN 978-3-540-60656-7. MR 0257325. Zbl 0176.00801.
  • Griffiths, Phillip; Harris, Joseph (1978). Principles of algebraic geometry. Pure and Applied Mathematics. New York: John Wiley & Sons. doi:10.1002/9781118032527. ISBN 0-471-32792-1. MR 0507725. Zbl 0408.14001.
  • Simon, Leon (1983). Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis. Vol. 3. Canberra: Centre for Mathematical Analysis at Australian National University. ISBN 0-86784-429-9. MR 0756417. Zbl 0546.49019.
  • Whitney, Hassler (1957). Geometric integration theory. Princeton Mathematical Series. Vol. 21. Princeton, NJ and London: Princeton University Press and Oxford University Press. doi:10.1515/9781400877577. ISBN 9780691652900. MR 0087148. Zbl 0083.28204..
  • Lin, Fanghua; Yang, Xiaoping (2003), Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston), vol. 1, Beijing/Boston: Science Press/International Press, pp. x+237, ISBN 978-1-57146-125-4, MR 2030862, Zbl 1074.49011

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