= Degree-zero correction to radial homotopy
{title2=$dh+hd=\mathrm{id}-F_0^*$}
For radial contraction on a star-shaped domain, the homotopy formula is the identity in positive form degrees because pullback by the constant map vanishes there. On functions it is $h_1df=f-f(0)$. A nonzero constant is a counterexample to the uncorrected degree-zero identity. The <Poincaré lemma> is therefore an exactness assertion in positive degrees, while closed degree-zero forms are locally constant.
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