The relationship between the support of cohomologous forms, possibly in the context of the Poincaré dual

de-rham-cohomologydifferential-geometryhomology-cohomologymanifolds

If we have $\omega = \omega' + d\tau$ for $\omega, \omega', \tau$ closed smooth $q$-forms of a smooth $n$-manifold, $1 \le q \le n-1$ and $\omega'$ has compact support, then does that mean $\omega$ has compact support?

Some thoughts

  1. I know that $\text{supp}(d\tau) = \text{supp}(-d\tau), \text{supp} \omega = \text{supp} \omega' \cup \text{supp}(d\tau)$ and $\text{supp} \omega' = \text{supp} \omega \cup \text{supp}(d\tau)$.

  2. There might be some algebra of sets rule I'm missing.

  3. I know that $\text{supp} \omega$ is compact if $\text{supp}(d\tau)$ is compact.

  4. I think $\gamma$ is in the kernel of $\int_{M}: Z^n_c M \to \mathbb R$ (see the prequel Section 24.1 for $Z$ and $B$) if and only if $\gamma \in B^n_c M$ by Theorem 10.13 in From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave, but I don't think we can choose $\gamma=d\tau$ because $\tau \in Z^q_cM, d\tau \in B^{q+1}_cM \subseteq Z^{q+1}_cM$, and I'm not sure we're given $q=n-1$.

Context:

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. I am trying to understand why "we can in fact demand the closed Poincaré dual of a compact oriented submanifold to have compact support." (Section 5)

My understanding is that

  1. for the closed Poincaré dual $\eta_S$ and compact Poincaré dual $\eta_S'$ of a compact $k$-submanifold $S$ of a smooth n-manifold $M$ and $H^{q}M = \frac{Z^{q}M}{B^{q}M}$ (see the prequel Section 24.1 for $Z$ and $B$), we have that $$\eta_S + B^{n-k}M = \eta_S' + B^{n-k}M$$ but we may have that $$(\eta_S + B^{n-k}M =) \ \eta_S' + B^{n-k}M \ne \eta_S' + B^{n-k}_cM$$

  2. Thus, what Bott and Tu mean by $\eta_S$ and $\eta_S'$ being the same as forms is that their equivalence classes modulo $B^{n-k}M$ are equal, but such equivalence class may be unequal to the equivalence class of $\eta_S'$ modulo $B^{n-k}_cM$.

  3. Thus, it's quite weird to strictly say that $\eta_S = \eta_S'$ because actually any $\tau_S \in \eta_S + B^{n-k}M$ (I'm avoiding the notation "$[\eta_S]$" because of the trauma this page has caused me) also fits the integral in (5.13) and similarly for $\tau_S' \in \eta_S' + B^{n-k}_cM$ and (5.14).

  4. Therefore, I interpret demanding the $\eta_S$ to have compact support is saying $\eta_S$, along with every $\tau_S \in \eta_S + B^{n-k}M$, has compact support because $\eta_S' \in \eta_S + B^{n-k}M$, suggesting there seems to be a hidden rule about the relationship of the supports of cohomologous forms.

Some more thoughts after context:

  1. Even if we can't integrate $\int_M d\tau$, maybe we can integrate $\int_S \iota^{*}(d\tau)$ for $\iota: S \to M$ inclusion and the pullback $\iota^{*}: \Omega^*M \to \Omega^*S$ inclusion to say support $\iota^*(d\tau) \subseteq$ support $d\tau \cap S$ is compact, but I don't know how this gives us support $d\tau$ is compact.

Best Answer

The confusion is probably caused by the authors, since they agree that

p.51 ... We will often call both the cohomology class $[\eta_S]$ and a form representing it the Poincare dual of $S$.

I suppose the same abuse is used for the compact Poincare dual.

In your point 2:

Thus, what Bott and Tu mean by $\eta_S$ and $\eta_S'$ being the same as forms is that their equivalence classes modulo $B^{n-k}M$ are equal, but such equivalence class may be unequal to the equivalence class of $\eta_S'$ modulo $B^{n-k}_cM$.

Indeed sometimes they explicitly say that $\eta_S$ and $\eta_S'$ are equal as forms. In this situation, they think of $\eta_S$ and $\eta_S'$ both as differential forms on $M$.

Indeed, $[\eta_S]$ and $[\eta_S']$ are elements in different spaces so it does not make sense to say that they are equal, apriori. But they did try to make sense of it, for example at p.51, they say

So as a form $\eta_S'$ is also the closed Poincare dual of $S$.

Then they try to make sense of the above sentence:

... ie, the natural map $H^{n-k}_c(M) \to H^{n-k} (M)$ sends the compact Poincare dual to the closed Poincare dual.

The natural map (called $i_*$) is induced by the inclusion $i : \Omega^{n-k}_c (M) \to \Omega^{n-k}(M)$ of differential forms. So they argue that $i_* [\eta_S']\in H^{n-k}(M)$ is the closed Poincare dual of $S$.

If the above is interpreted as forms (instead of cohomology class), then $\eta_S'$ as a compactly supported differential form in $M$, is also the closed Poincare dual of $M$.

As a result, they claim that the closed Poincare dual of $S$ is also represented by a compactly supported differential form (which is $\eta_S'$). So by compactly supported Poincare dual, they mean that there is a compactly supported differential form inside the class. It does not mean that all elements in that class are of compact support.

So if $\omega = \omega' + d\tau$ and $\omega'$ is of compact support, then $\omega$ might not have compact support.

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