Closed Poincaré dual is $\int_M \eta_S \wedge \omega $ and not $\int_M \omega \wedge \eta_S$. What about the compact Poincaré dual?

My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel.

The characterization of the closed Poincaré dual is given here (the “(5.13)”) in Section 5.5. This has $ \int_M \omega \wedge \eta_S$ , where $ \eta_S$ is on the right rather than left. The comments here confirm that this is a mistake.

I believe the characterization for the compact Poincaré dual for compact $ S$ and $ M$ of finite type given here (the “(5.14)”) is correct with $ \eta_S’$ on the right. But since the “(5.13)” is a mistake, I think this may have some implications for the compact Poincaré dual.

Bott and Tu say

If (5.14) holds for any closed k-form $ \omega$ , then it certainly holds for any closed k-form $ \omega$ with compact support. So as a form, $ \eta_S’$ is also the closed Poincaré dual, i.e., the natural map $ H^{n-k}_c(M) \to H^{n-k} (M)$ sends the compact Poincaré dual to the closed Poincaré dual.

I think the quote implicitly references “(5.13)”. My question depends whether or not such thought is correct:

  1. If the quote implicitly references “(5.13)”, then what does the mistake of (5.13) mean for the claim that the compact Poincaré dual is also the closed Poincaré dual?

    • A. The claim is still correct and still for the same reason.

    • B. The claim is still correct but for a different reason.

    • C. The claim is now incorrect.

  2. If the quote does not implicitly reference “(5.13)”, then I’ve apparently misunderstood. How do Bott and Tu say compact Poincaré dual is also the closed Poincaré dual?

Closed Poincaré dual, why $\int_M \omega \wedge \eta_S$ and not $\int_M \eta_S \wedge \omega $?

My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel.

The characterization of the closed Poincaré dual is given here (the “(5.13)”) in Section 5.5. This has $ \int_M \omega \wedge \eta_S$ , where $ \eta_S$ is on the right rather than left.

Question: Why is it $ \int_M \omega \wedge \eta_S$ , where $ \eta_S$ is on the right rather than left?

  • See below for why I think $ \eta_S$ should be on the left rather than right.

  • Previously, in Section 5.3, we had this equivalent definition (the “Lemma”) of a nondegenerate pairing between two finite-dimensional vector spaces and Poincaré duality (the “(5.4)”).

  • Note: I believe the characterization for compact Poincaré dual for compact $ S$ and $ M$ of finite type is correct with $ \eta_S’$ on the right.

  • Guess: Could have something to do with sign commutativity of Mayer-Vietoris, as described in Lemma 5.6.

  • Guess: Poincare dual as described is indeed with $ \eta_S$ on the left, but there’s also a unique cohomology class $ [\gamma_S]$ that’s on the right given by $ [\gamma_S] = [-\eta_S]$ .


How I got $ \int_M \eta_S \wedge \omega$ instead of $ \int_M \omega \wedge \eta_S$ :

I use $ ()^{\vee}$ , instead of $ ()^{*}$ , to denote dual just like in Section 3.1 of the prequel.

  1. Let $ \varphi$ be the “linear functional on $ H^{k}_cM$ ” given here.

    • Such $ \varphi: H^{k}_cM \to \mathbb R$ is given by $ \varphi[\omega] = \int_S \iota^{*}\omega$ for $ [\omega] \in H^k_cM$ and $ \iota: S \to M$ inclusion.
  2. Let $ \delta$ be the isomorphism of Poincaré duality (the “(5.4)”).

    • Such $ \delta: H^{n-k}M \to (H^{k}_cM)^{\vee}$ is given by $ \delta([\tau]) = \delta_{[\tau]}$ , for $ [\tau] \in H^{n-k}M$ and $ \delta_{[\tau]}$ given below.

    • $ \delta_{[\tau]}([\omega]) = \int_M (\tau \wedge \omega)$ , for $ [\omega] \in H^k_cM$ , under the well-definedness described in Section 24.4 of the prequel (which I think is the full details of the “Because the wedge product is an antiderivation, it descends to cohomology” here) and under the pairing given here, which I believe puts $ \tau$ on the left rather than right.

  3. $ [\eta_S]$ is the inverse image of $ \varphi$ under $ \delta$ .

    • By choosing $ [\tau] = [\eta_S]$ , we get $ \delta([\eta_S]) = \delta_{[\eta_S]} = \varphi$ , that is, for all $ [\omega] \in H^k_cM$ ,

$ $ \int_M (\eta_S \wedge \omega) = \int_S \iota^{*}\omega,$ $

where $ \eta_S$ is on the left rather than right.