coboundary definition of exterior derivative

Let $M$ be a smooth manifold, and

• let $C^{\mathrm{\infty }}(M)$ denote the algebra of smooth functions on $M$;

• let $V(M)$ denote the Lie-algebra of smooth vector fields;

• and let ${\mathrm{\Omega }}^k(M)$ denote the vector space of smooth,differential $k$-forms.

Recall that a differential form $\alpha \in {\mathrm{\Omega }}^k(M)$ is amultilinear, alternating mapping

$$\alpha :V(M)\times \mathrm{\cdots }\times V(M)\text{(k times)}\to C^{\mathrm{\infty }}(M)$$

such that, in local coordinates,$\alpha$ looks like a multilinear combination of its vector fieldarguments. Thus, employing the Einstein summation convention and localcoordinates , we have

$$\alpha (u,v,\mathrm{\dots },w)={\alpha }_{ij\mathrm{\dots }k}{u}^i{v}^j\mathrm{\cdots }{w}^k,$$

where $u,v,\mathrm{\dots },w$ is a list of $k$ vector fields. Recall alsothat $C^{\mathrm{\infty }}(M)$ is a $V(M)$ module. The action is given bya directional derivative, and takes the form

$$v(f)=v^i{\partial }_{i}f,v\in V(M),f\in C^{\mathrm{\infty }}(M).$$

With these preliminaries out of the way, we have the followingdescription of the exterior derivative operator $d:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k+1}(M)$. For $\omega \in {\mathrm{\Omega }}^k(M)$, we have

$(d\omega )(v_0,v_1,\mathrm{\dots },v_k)=$${\displaystyle \sum _{0\le i\le k}}{(-1)}^k{v}_i\omega (\mathrm{\dots },{\widehat{v}}_i,\mathrm{\dots })+$(1)

where ${\widehat{v}}_i$ indicates the omission of the argument $v_i$.

The above expression (1) of $d\omega$ can be taken asthe definition of the exterior derivative. Letting the $v_i$ arguments be coordinate vector fields, it is not hard to show that the above definition is equivalent to theusual definition of $d$ as a derivation of the exterior algebra ofdifferential forms, or the local coordinate definition of $d$. Thenice feature of (1) is that it is equivalent to thedefinition of the coboundary operator for Lie algebra cohomology.Thus, we see that de Rham cohomology, which is the cohomology of thecochain complex $d:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k+1}(M)$, is justzeroth-order Lie algebra cohomology of $V(M)$ with coefficients in$C^{\mathrm{\infty }}(M)$. The bit about “zeroth order” means that we areconsidering cochains that are zeroth order differential operators oftheir arguments — in other words, differential forms.