# Poincaré Duality for non-compact manifolds without Zorn's Lemma

Does exists a proof of the Poincaré Duality version for non-compact manifolds without using the Zorn's Lemma? I know that there is a proof using the Whitney embedding theorem, but I don't know this theorem's proof, so I don't know if it uses the Zorn's Lemma too.

• Where is Zorn's lemma used in the proof you have in mind? – Qiaochu Yuan May 13 '16 at 7:20
• Sorry, mistake, editing. – MonsieurGalois May 13 '16 at 7:25
• I meant the Poincaré Duality for non-compact manifolds. The proof I read is the one from Hatcher. – MonsieurGalois May 13 '16 at 7:27
• Yes, Poincare's original proof works just fine -- for manifolds with triangulations. Technically, Whitehead's triangulations proof uses Whitney's theorem but triangulability is quite different in character. Whitney's theorem has fairly direct proof. I encourage you (really, everyone) to learn the proof. It is analogous to quite a few other basic theorems. I suppose the handlebody proof of Poincare duality does not use Zorn's lemma in any heavy-handed way. – Ryan Budney May 13 '16 at 7:34
• Can you add the proof you're telling me to read it please? – MonsieurGalois May 13 '16 at 7:37

A slight modification of Hatchers Proof gives you what you want: Hatcher relies on the fact that if $M$ is a union of opens totally ordered by inclusion, which satisfy PD, then $M$ does, too. It is easy to see that you can weaken that to the opens only being directed. Also, another case tells you that all finite intersections of coordinate balls satisfy PD, so all finite unions do too. Then, at the end instead of using Zorn's Lemma you only have to look at the cover of all finite unions of coordinate balls, which are obviously directed w.r.t. inclusion.