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stub for Poincare lemma
I have added a little bit of substance to Poincare lemma
@Anyone - the flow of statements at Poincare lemma didn’t make sense to me, so I popped a note in. I made a small correction to a typo (manifolds had changed names mysteriously).
Phew, I hadn’t been concentrating when typing. Thanks for catching this.
Where it said “$H_{dR}^\bullet(X) = H_{dR}^\bullet(Y)$” is was supposed to say “$H_{dR}^\bullet(f_1^*) = H_{dR}^\bullet(f_2^*)$”. Namely:
if two cochain maps $f_1^*$ and $f_2^*$ are homotopic, then they induce the same maps in cohomology.
Sorry. I have fixed it now. Any further typos? I need to call it quits now.
I have added more details. (Hopefully without weird typos this time!)
I have expanded the Idea-section at Poincare lemma a bit, mentioning also the complex-analytic version and the incarnation of the statement as the local quasi-iso $\mathbb{R}\to \Omega^\bullet_{dR}$ ($\mathbb{C}\to \Omega^\bullet_{hol}$). Wanted to do more, but am out of steam now.
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