title: Homological integration
text: # 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 ∈ $Ω^k.$ This is a homological rather than cohomological construction.
  
  