Radial homotopy operator ($H\alpha ={\int }_0^1{\rho }_t^{\ast }({\iota }_R\alpha )dt/t$)

On a star-shaped domain, let ${\rho }_t(x)=tx$ and $R=\sum x_j{\partial }_{x_j}$. On positive-degree differential forms, $H\alpha ={\int }_0^1{\rho }_t^{\ast }({\iota }_R\alpha )dt/t$ satisfies $dH+Hd=\mathrm{id}-{\rho }_0^{\ast }$. It gives a primitive of every closed positive-degree form. On a star-shaped domain in a complex vector space, contraction with the two type components of $R$ shows that $H$ sends type $(p,q)$ into $(p-1,q)\oplus (p,q-1)$. There is no primitive assertion for a nonzero constant function.

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 $h_{1}df=f-f(0)$. 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.

On a fiberwise star-shaped neighborhood, let $H_s$ scale fibers by $s$ and let $E$ be the radial vector field. For a closed differential form $\delta$ with zero pullback to the zero section, the homotopy primitive $\eta ={\int }_0^1{s}^{-1}{H}_s^{\ast }({\iota }_E\delta )ds$ satisfies $d\eta =\delta$. For a two-form, it vanishes as an ambient covector along that section, providing the fixed-submanifold condition in the relative Moser theorem.