Editing Chain complex (section)

===de Rham cohomology===
{{main|de Rham cohomology}}
The [[differential form|differential ''k''-forms]] on any [[smooth manifold]] ''M'' form a [[real number|real]] [[vector space]] called Ω<sup>''k''</sup>(''M'') under addition. 
The [[exterior derivative]] ''d'' maps Ω<sup>''k''</sup>(''M'') to Ω<sup>''k''+1</sup>(''M''), and ''d''{{i sup|2}} = 0 follows essentially from [[symmetry of second derivatives]], so the vector spaces of ''k''-forms along with the exterior derivative are a cochain complex.

:<math> 0\stackrel{\subset}{\to}\ {\Re^{c}}  \stackrel{\subset}{\to}\ {\Omega^0(M)} \stackrel{d}{\to}\ {\Omega^1(M)} \stackrel{d}{\to}\ {\Omega^2(M)} \stackrel{d}{\to}\ \Omega^3(M) \to \cdots</math>

The cohomology of this complex is called the '''de Rham cohomology''' of ''M''. [[Locally constant function|Locally constant functions]] are designated with its isomorphism <math> \Re^c</math> with c the count of mutually disconnected components of ''M''. This way the complex was extended to leave the complex exact at zero-form level using the subset operator.

[[Smoothness#Smooth functions on and between manifolds|Smooth maps]] between manifolds induce chain maps, and smooth homotopies between maps induce chain homotopies.