De Rham integration cochain

Definition [F1] gives one uniquely determined value on every smooth $k$-simplex. For each finite chain, distributing its coefficients in the displayed sum shows that this function extends to a real linear functional. Two such functionals agreeing on all simplices agree on every finite chain. By [F2] this defines a unique element $I_M^k(\omega )$ of the cochain space. No choice of a vector-space basis or of simplex extensions is made.

For forms $\omega ,\eta$ of degree $k$ and real $a,b$, alternating evaluation in the definition of pullback gives ${\sigma }^{\ast }(a\omega +b\eta )=a{\sigma }^{\ast }\omega +b{\sigma }^{\ast }\eta$. For $k>0$, the Jordan coefficient integral is linear by [F4]; for $k=0$, evaluation is linear directly. Thus $I_M^k(a\omega +b\eta )(\sigma )=a{I}_M^k(\omega )(\sigma )+b{I}_M^k(\eta )(\sigma )$ on every simplex, and step 1.1 makes $I_M^k$ a real linear map.

For $k=0$ the value at a point simplex is exactly the function value there; for $k=1$ it is the integral of the pulled-back one-form over the closed oriented interval. Degenerate simplices remain in the supplied set and receive their actual integrals, rather than being removed. If $k>\dim M$, [F3] gives zero source and the map is zero, even though the smooth cochain space may be nonzero. Negative degrees are zero on both sides. On an empty manifold there are no simplices and the only cochain is zero. These conventions include the zero form and zero chain and require no choice.