Finding generators for cohomology group by analysing Mayer-Vietoris sequence

Question

I am working through my second book on the topic of smooth manifolds, and like the first time, it was all very much within my grasp, right up until the cohomology chapter(s). I am getting the impression that I am fundamentally missing something, and I could not find an example online that really illustrates an approach to this without invoking some prior knowledge about generators for the cohomology groups of particular spaces. I find this genuinely frustrating because I really do want to understanding this subject, so I am hoping that someone can show me how to deal with this particular example, in the hopes that I will learn how to approach such problems in general.
The exact problem I am currently working on is 17-5 in Lee's ISM. It reads: for each $n\geq 1$, compute the de Rham cohomology groups of $M:=\mathbb{R}^n-\{e_1,-e_1\}$, and for each non-zero cohomology group, give explicit generators. For $n=1$, the problem is easy, we just get three components that are diffeomorphic to the real line itself. Assume $n>1$. I approached the problem as follows. Let $U$ and $V$ be open sets containing $e_1$ and $-e_1$ respectively, such that their intersection is contractible to a point, and each of $U$ and $V$ is diffeomorphic to $\mathbb{R}^n-\{0\}$ (e.g. $U=(-\infty,1/2)\times \mathbb{R}^{n-1}-\{-e_1\}$ and $V=(-1/2,\infty)\times\mathbb{R}^{n-1}-\{e_1\}$), for which the book already showed that $$H^p(\mathbb{R}^n-\{0\})\begin{cases} \mathbb{R}& \text{if } p=0,n-1\\ 0 & \text{otherwise} \end{cases} $$ Thus, we get the sequence $$ 0\rightarrow H^{n-1}(M)\rightarrow H^{n-1}(U)\oplus H^{n-1}(V)\rightarrow H^{n-1}(U\cap V)\rightarrow H^n(M)\rightarrow 0 $$ And substituting the known elements of this sequence, it becomes $$ 0\rightarrow H^{n-1}(M)\rightarrow \mathbb{R}\oplus \mathbb{R}\rightarrow 0\rightarrow H^n(M)\rightarrow 0 $$ from which I deduced that $H^n(M)=0$, whereas $H^{n-1}(M)=\mathbb{R}^2$. I might have made a mistake up until this point, in which case, please correct me.
The next part is where I really lose track of what I am actually supposed to do; constructing the generator(s). How would I do this? My attempt was somewhat miserable (I did not get far at all), but let me share it anyway. We have the maps $$ k^*\oplus l^*:H^p(M)\to H^p(U)\oplus H^p(V)\\ i^*-j^*:H^p(U)\oplus H^p(V)\to H^p(U\cap V)\\ \delta:H^p(U\cap V)\to H^{p+1}(M) $$ Defined by $$(k^*\oplus l^*)\omega=(\omega|_U,\omega|_V)\quad (i^*-j^*)(\omega,\eta)=\omega|_{U\cap V}-\eta|_{U\cap V}$$ and $\delta$ is the connecting homomorphism. Then, an element $(u,v)\in \mathbb{R}\oplus\mathbb{R}$ represents a form $u$ on $U$ and $v$ on $V$. Thus $(i^*-j^*)(u,v)=u-v$. I do not know if this is even relevant information, or indeed how to proceed. I would very much appreciate a detailed explanation on how to do this, but a simpler answer that merely hints at the right direction would also be helpful.
EDIT: I showed in exercise 16-9 of the same book that $\omega=|x|^{-n}\sum(-1)^{i-1}x^idx^1\wedge\dots\wedge\widehat{dx^i}\wedge\dots\wedge dx^n$ is a smooth closed $n-1$ form on $\mathbb{R}^n-\{0\}$ that is not exact. This is the form that @Kajelad was talking about in the comments also. Can I "patch this together" by taking two of these forms, shifting the singularity to $e_1$ and $-e_1$ respectively, and then restricting them by a partition of unity subordinate to the cover $\{U,V\}$? Or ought they just be pulled back along the inclusion map? If not, I am still lost. If it does work, I am not completely sure why this works, so I would still appreciate an explanation.

Answer 1

It's possible to do things more or less as you describe, but you can't simply multiply the differential forms by a partition of unity. Your Mayer-Vietoris sequence contains an isomorphism $$ \Theta:H^{n-1}M\to H^{n-1}U\oplus H^{n-1}V \\ \Theta([\omega])=[\omega|_U],[\omega|_V] $$ Choosing a partition of unity $\psi_U,\psi_V$ subbodinate to $\{U,V\}$, $\Theta^{-1}$ can be written in terms of representatives by $$ \Theta^{-1}([\mu],[\nu])=[\psi_U\mu+\psi_V\nu+d\psi_U\wedge\lambda] $$ Where $\lambda\in\Omega^{n-2}(U\cap V)$ satisfies $d\lambda=\mu|_{U\cap V}-\nu|_{U\cap V}$. In this case, though, the equality above is not easy to use, since we would need to find an explicit expression for $\lambda$ in coordinates.

Here's a more convenient way of obtaining the generators which doesn't require any partitions of unity. Consider a slightly different set of inclusions: $$ \mathbb{R}^n\setminus\{e_1,-e_1\} \\ \swarrow\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \searrow\ \ \ \\ \mathbb{R}^n\setminus\{e_1\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathbb{R}^n\setminus\{-e_1\} \\ \searrow\ \ \ \ \ \ \swarrow\ \ \ \\ \mathbb{R}^n\ \ \ \ $$ These gives rise to a MVS of the form $$ \vdots \\ H^{n-1}\mathbb{R}^n=0\\ \downarrow \\ H^{n-1}(\mathbb{R}^n\setminus\{e_1\})\oplus H^{n-1}(\mathbb{R}^n\setminus\{-e_1\}) \\ \downarrow\Phi \\ H^{n-1}(\mathbb{R}^n\setminus\{e_1,-e_1\}) \\ \downarrow \\ H^n\mathbb{R}^n=0 \\ \vdots $$ Where $\Phi$ is an isomorphism, given by $$ \Phi([\mu],[\nu])=\left[\mu|_{\mathbb{R}^n\setminus\{e_1,-e_1\}}-\nu|_{\mathbb{R}^n\setminus\{e_1,-e_1\}}\right] $$ Since you already have a generator for $H^{n-1}(\mathbb{R}^n\setminus\{0\})$, you can obtain generators for $H^{n-1}(\mathbb{R}^n\setminus\{\pm e_1\})$ simply by translation, and the resulting forms restrict to generators for $H^{n-1}(\mathbb{R}^n\setminus\{e_1,-e_1\})$ via $\Phi$.