I have copied below exercise 4.8.7 from Guillemin and Pollack's Differential Topology
I'm having trouble following the last part of the hint (Exercise 7 of Section 6 refers to the fact that homotopic maps induce the same map on cohomology).
Following the hint, we pick a form $\theta \in \Omega^k(X)$ and an open cover $\{U_1,\ldots,U_n\}$ where $U_i$ is smoothly isotoptic to a coordinate neighbourhood we started with $U$. Then we choose a partition of unity subordinate to this cover i.e. we have smooth functions $\rho_i$'s with $\mathrm{supp}(\rho_i) \subset U_i$ and $\sum_{i=1}^n\rho_i \equiv 1$. Letting $\theta_i = \rho_i\theta$, we have forms supported in $U_i$ such that they add upto to $\theta$. Okay, now, since $U_i$ is smooth isotopic to $U$, we have a isomorphism of cohomology groups $H^k(U_i) \cong H^i(U)$, so $[\theta_i]$ corresponds to some class of a scalar multiple of $[\omega]$. But the hint is saying that we make it so that $\theta_i$ is cohomologous to a scalar multiple of $\omega$, that makes me think I'm not using the fact that $U_i$ and $U$ are isotopic well enough. I suppose I'm also confused on how're we intepreting isotopy in this context.
Any comments will be helpful. I know I'm asking for help in the middle a chain of thought that you may not be privy to, so apologies for that!
Update: After following the comments, we have the following. Let $h_{i,t}$ denote the isotopy between $U$ and $U_i$, then for $h_{i,1}^*\theta_i$, as a compactly supported form on $U$, we have $[h_{i,1}^*\theta_i] = c_i[\omega]$, where $c_i = \int_Uh_{i,1}^*\theta_i = \int_{U_i}\theta_i$, since $h_{i,1}^*$ is a diffeomorphism (this comes from the initial hint and 4.8.6).
Then I suppose letting $H_i$ denote the inverse of $h_{i,1}$, we have $[\theta_i] = c_i[H_i^*\omega]$ and so $[\theta] = \sum_{i=1}^nc_i[H_i^*\omega]$.
Best Answer
The key to this problem is how we construct this cover given in the hint, which I initially took as a black box. We started with a $U$, let's fix a $p\in U$. Then for any $x\in X$, by the Isotopy lemma, we can find a diffeomorphism $h_x:X \to X$ which is isotopic to the identity on $X$ such that $h_x(p) = x$. Then $\{h_x(U)\}_{x\in X}$ is an open cover of a compact manifold, so we can choose a finite subcover, say $\{h_1(U),\ldots,h_N(U)\}$. Then $U_i = h_i(U)$ is the cover given in the hint. Since $h_i \sim \mathrm{id}_X$, then $h_i^\sharp = \mathrm{id}_X^\sharp$ (pullback maps on cohomology), and things follow. I won't bother fixing notation in my question and the update I added. But everything comes together to give us a solution. (I refer to and use notations from Guillemin-Pollack.)