My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. It seems to be claimed

  1. In the proof of Lemma 10.17:

For every proper open subset $W$ of $\mathbb R^n$, there is an $\omega \in \Omega_c^{n}(\mathbb R^n)$, namely the "$\omega_1$" in the proof, where $\int_{\mathbb R^n} \omega = 1$ and $\text{supp} \ \omega \subseteq W$.

  1. In the proof of Theorem 11.9:

For a compact connected oriented smooth n-manifold $M$ and for every open subset $U$ of $M$, there is an $\omega \in \Omega_c^{n}(M) = \Omega^{n}(M)$, where $\int_M \omega = 1$ and $\text{supp} \ \omega \subseteq U$ (I don't believe this is dependent on the particular $U$ from Lemma 11.8).

Question: Why?

Please try to answer using the tools in the book such as Theorem 10.13 (or Corollary 10.14), Lemma 10.15, the last sentence of this or Lemma 10.3(ii).

  • I think I'm missing something obvious about how such $\omega$ exists because the authors state it so naturally.

  • Is the fact of the existence of such $\omega$ actually not obvious to the reader at this point in the text? For non-obvious facts, I think authors would usually say something like "First/next, observe that (insert fact), the proof of which is left to the reader/to the exercises".

Here are my thoughts (assuming I understand right):

  1. I think (2) follows from Theorem 10.13 (or Corollary 10.14).

    • I get $\tau \in \Omega_c^{n}(M)$ , where $\int_{M} \tau = 1$ and $\text{supp} \ \tau \subseteq M$. Choose $\omega$ to be the zero extension of $\tau|_{U}$ to $\tilde{\tau|_{U}} = \tau|_{U}1_{U} + 0 \ 1_{U^c}$. Similarly, (1) would follow from Lemma 10.15: I get $\tau \in \Omega_c^{n}(\mathbb R^n)$, $\int_{\mathbb R^n} \tau = 1$, $\text{supp} \ \tau \subseteq \mathbb R^n$, and then $\omega = \tilde{\tau|_{W}} = \tau|_{W}1_{W} + 0 \ 1_{W^c}$.
  2. For (2), instead of applying Theorem 10.13 to $M$, I'll apply Theorem 10.13 to $U$, where the proof of Theorem 11.9 says we may actually be assume $U$ to be connected.

    • We get $\gamma \in \Omega_c^n(U)$, where $\int_{U} \gamma = 1$ and $\text{supp} \ \gamma \subseteq U$. Let $\psi = \gamma|_{\text{supp} \gamma}$, the restriction of $\gamma$ to its support. Denote the zero extensions of $\gamma$ and $\psi$ as $\tilde{\gamma} = \gamma 1_{U} + 0 \ 1_{M \setminus U}$ and $\tilde{\psi} = \psi 1_{\text{supp} \gamma} + 0 \ 1_{M \setminus \text{supp} \gamma}$.

    • Observe $\tilde{\psi} = \tilde{\gamma}$ (like here), and both are smooth by the last sentence of this.

    • Choose $\omega = \tilde{\psi} = \tilde{\gamma}$: As $\omega = \tilde{\psi}$, $\omega$ will satisfy $\text{supp} \ \omega \subseteq U$. As $\omega = \tilde{\gamma}$, $\omega$ will satisfy $\int_{M} \omega = 1$ because $\int_{M} \tilde{\gamma} = \int_{U} \gamma$ by Lemma 10.3(ii).

    • But with this method, how do we argue similarly for (1), where the proof of Lemma 10.17 does not say that we may assume $W$ is connected?


An open subset contain the domain of a chart $g:U\rightarrow\mathbb{R}^n$ which is diffeomorphic to an open subset of $\mathbb{R}^n$, let $f$ be a cut-off function defined on $g(U)$, (the support of $f$ is contained in $g(U)$, $f\neq 0$) and $\omega_0$ the standard volume form of $\mathbb{R}^n$. There exists an $n$-form $\omega$ defined on $M$ whose restriction to $U$ is $g^*(f\omega_0)$ and the restriction of $\omega$ on $M-U$ is zero. Define $\omega'={1\over{\int_M\omega}}\omega$.

  • $\begingroup$ Thanks. Kinda get the gist but working toward understanding details. 1. Is there something wrong with what I tried to do? I mean is Lemma 10.15 (for Lemma 10.17) or Theorem 10.13 (for Theorem 11.9) irrelevant? 2. How is cut-off related to bump please? By bump, I refer to Tu's book. Here's the definition... $\endgroup$ – Selene Auckland May 2 at 8:54
  • $\begingroup$ ...and an exercise, where I think your $f$ comes from. (Not sure if same bump as in wikipedia.) In particular, if your $f$ does come from the exercise, then I better understand the role of $W$ nonempty for Lemma 10.17. $\endgroup$ – Selene Auckland May 2 at 8:54
  • $\begingroup$ Tsemo Aristide, is $(g,U)$ a positively oriented chart or at least an oriented chart? See my other question please $\endgroup$ – Selene Auckland May 2 at 10:28
  • $\begingroup$ Tsemo Aristide, I have made my post much shorter hopefully. Could you please comment on my attempts? $\endgroup$ – Selene Auckland May 13 at 4:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.