The local de Rham comparison on a ball

The radial homotopy is $H(x,t)=c+t(x-c)$. For a $k$-form $\omega$ with $k\ge 1$, its homotopy operator is explicitly $(L_H\omega {)}_x(v_1,\dots ,v_{k-1})={\int }_0^1{t}^{k-1}{\omega }_{c+t(x-c)}(x-c,v_1,\dots ,v_{k-1})dt\mathrm{.}$ The homotopy identity in [F1] gives $d{L}_H\omega +L_{H}d\omega =\omega -H_0^{\ast }\omega$; since the constant-map pullback $H_0^{\ast }\omega$ is zero in positive degree, every closed positive-degree form is exact. A closed zero-form satisfies $f(x)=f(c)$, so evaluation at $c$ identifies $H_{\mathrm{d}\mathrm{R}}^0(B)$ with $\mathbb{R}$.

On a smooth singular $q$-simplex $\sigma$, triangulate ${\mathrm{\Delta }}^q\times [0,1]$ by the $q+1$ affine prism simplices ${\kappa }_i$ with ordered vertices $(v_0,0),\dots ,(v_i,0),(v_i,1),\dots ,(v_q,1),$ and put $P\sigma ={\sum }_{i=0}^q(-1{)}^{i}H(\sigma \times \mathrm{id}){\kappa }_i$. The cancellations of paired interior faces leave $\mathrm{\partial }P+P\mathrm{\partial }={\mathrm{id}}_{\mathrm{\#}}-(c{)}_{\mathrm{\#}}\mathrm{.}$ Dualizing gives $\mathrm{id}-c^{\ast }=\delta D+D\delta$, not a contraction of the unnormalized complex by itself: positive-dimensional constant simplices remain nonzero. The point complex is the alternating complex with one generator in every degree, whose positive cohomology vanishes, and [F1] uses this calculation together with the displayed homotopy to prove that inclusion of the centre and constant projection induce inverse cohomology maps. Hence $H_{\mathrm{\infty }}^0(B;\mathbb{R})=\mathbb{R}$ and the positive groups vanish.

Integration sends a constant function $a$ to the zero-cochain taking value $a$ at every vertex. Hence it is the identity under the two degree-zero evaluations, and in positive degrees it is the unique map $0\to 0$. If $n=0$, the ball is a point and the unnormalized constant simplices still contract as in [F1]. Zero forms, constant simplices, and both homotopy endpoints are included above. One centre and explicit finite prism sums are used, so no choice principle enters.