Editing Hodge star operator (section)

==== Poincare lemma for codifferential ====
In analogy to the [[Poincare lemma]] for [[exterior derivative]], one can define its version for codifferential, which reads<ref name=":0">{{Cite journal |last=Kycia |first=Radosław Antoni |date=2022-07-29 |title=The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications to Physics |url=https://doi.org/10.1007/s00025-022-01646-z |journal=Results in Mathematics |language=en |volume=77 |issue=5 |pages=182 |doi=10.1007/s00025-022-01646-z |issn=1420-9012|arxiv=2009.08542 |s2cid=221802588 }}</ref>
: ''If'' <math>\delta\omega=0</math> ''for'' <math>\omega \in \Lambda^{k}(U)</math>'', where '' <math>U</math> ''is a [[star domain]] on a manifold, then there is'' <math>\alpha \in \Lambda^{k+1}(U)</math> ''such that'' <math>\omega=\delta\alpha</math>''.''

A practical way of finding <math>\alpha</math> is to use cohomotopy operator <math>h</math>, that is a local inverse of <math>\delta</math>. One has to define a [[homotopy operator]]<ref name=":0" />
: <math>H\beta = \int_{0}^{1} \mathcal{K}\lrcorner\beta|_{F(t,x)}t^{k}dt,</math> 
where <math>F(t,x)=x_{0}+t(x-x_{0})</math> is the linear homotopy between its center <math>x_{0}\in U</math> and a point <math>x \in U</math>, and the (Euler) vector <math>\mathcal{K}=\sum_{i=1}^{n}(x-x_{0})^{i}\partial_{x^{i}}</math> for <math>n=\dim(U)</math> is inserted into the form <math>\beta \in \Lambda^{*}(U)</math>. We can then define cohomotopy operator as<ref name=":0" />
: <math>h:\Lambda(U)\rightarrow \Lambda(U), \quad h:=\eta {\star}^{-1}H\star</math>,
where <math>\eta \beta = (-1)^{k}\beta</math> for <math>\beta \in \Lambda^{k}(U)</math>.

The cohomotopy operator fulfills (co)homotopy invariance formula<ref name=":0" /> 
: <math>\delta h + h\delta = I - S_{x_{0}} ,</math>
where <math>S_{x_{0}}={\star}^{-1}s_{x_{0}}^{*}{\star}</math>, and <math>s_{x_{0}}^{*}</math> is the [[Pullback (differential geometry)|pullback]] along the constant map <math>s_{x_{0}}:x \rightarrow x_{0}</math>.

Therefore, if we want to solve the equation <math>\delta \omega =0</math>, applying cohomotopy invariance formula we get
: <math> \omega= \delta h\omega + S_{x_{0}}\omega,</math> where <math>h\omega\in \Lambda^{k+1}(U)</math> is a differential form we are looking for, and "constant of integration" <math>S_{x_{0}}\omega</math> vanishes unless <math>\omega</math> is a top form.

Cohomotopy operator fulfills the following properties:<ref name=":0" /> <math>h^{2}=0, \quad \delta h \delta =\delta, \quad h\delta h =h</math>. They make it possible to use it to define<ref name=":0" /> ''anticoexact'' forms on <math>U</math> by <math>\mathcal{Y}(U)=\{ \omega\in\Lambda(U)| \omega = h\delta \omega \}</math>, which together with [[Exact form|exact forms]] <math>\mathcal{C}(U) =\{ \omega\in\Lambda(U)|\omega = \delta h\omega \}</math> make a [[direct sum]] decomposition<ref name=":0" /> 
: <math>\Lambda(U)=\mathcal{C}(U)\oplus \mathcal{Y}(U)</math>.

This direct sum is another way of saying that the cohomotopy invariance formula is a decomposition of unity, and the [[Projector operator|projector operators]] on the summands fulfills [[Idempotent (ring theory)|idempotence]] formulas:<ref name=":0" /> <math>(h\delta)^{2}=h\delta, \quad (\delta h)^{2}=\delta h</math>.

These results are extension of similar results for exterior derivative.<ref>{{Cite book |last=Edelen |first=Dominic G. B. |url=https://www.worldcat.org/oclc/56347718 |title=Applied exterior calculus |date=2005 |isbn=978-0-486-43871-9 |edition=Revised |location=Mineola, N.Y. |oclc=56347718}}</ref>