My book has a sketch of what a proof of Poincare's lemma looks like:
Definitions:
Let $U\subset \mathbb{R}^m$ open. Represent a point of $U\times \mathbb{R}$ by $(x,t) = (x_1,\cdots, x_m, t)$. Every form $w$, of degree $r$ in $U\times \mathbb{R}$, is written in a unique way as $w = dt\wedge \alpha + \beta$, where $\alpha(x,t) = \sum_{I}\alpha_I(x,t)d_{x_I}$ and $\beta(x,t) = \sum_J b_J(x,t)d_{x_j}$ are forms of degree $r-1$ and $r$ respectively.
I understood the deinition, but why $w = dt\wedge \alpha + \beta$?
Define a linear map $K$, which associates to each continuous form of degree $r$ in $U\times\mathbb{R}$, a continuous form of degree $r-1$ in $U$, by doing $(Kw)(x) = \int_0^1 \alpha(x,t)\ dt = \sum_I(\int_0^1 \alpha_I(x,t)\ dt)\ dx_I$. For each $t\in\mathbb{R}$, consider the application $i_t:U\to U\times\mathbb{R}$, defined by $i_t(x) = (x,t)$.
I understood the definitions, they're just one variable integrals, and that $ \int_0^1 \alpha(x,t)\ dt = \sum_I(\int_0^1\cdots)dx_I$ comes entirely from the definition of $\alpha$. But what am I supposed to do with the application $i_t$? I guess it's just gonna be useful in the next step. For now I just have to consider it, right?
Now we just have to prove the following:
Prove that, for each form $w$, of class $C^1$ in $U\times\mathbb{R}$, we have $K(dw) + d(Kw) = i_1^*(w)-i_0(w)$. Finish by seeing that, at each homotopy $H:U\times[0,1]\to V\subset\mathbb{R}^n$, of class $C^{\infty}$, between applications $f,g:U\to V$, we can associate a linear map $L = K\circ H^*$, which takes forms of degree $r$ in $V$ over forms of degree $r-1$ in $U$, in such a way that $L(dw) + d(Lw) = g^*w -f^*w$, for all forms $w$, of class $C^{\infty}$ in $V$. From this, if $w$ is closed, then the difference $g^*w-f^*w$ is exact. In particular, if the open $U$ is $C^{\infty}$ contractible, then every form of degree $>0$ in $U$ is exact. More precisely, this occrus in a convex $U$, in special, with $\mathbb{R}^m$
I'm trying to extend $H$ to an application $\overline{H}:U\times \mathbb{R}\to V$ by doing $\overline{H}(x,t) = H(x,\xi(t))$, with $0\le \xi(t)\le 1, \xi(t) = 0$ if $t\le 0$ and $\xi(t) = 1$ if $t\ge 1$.
I'm trying to follow this proof: http://math.columbia.edu/~dlitt/exposnotes/poincare_lemma.pdf but it's quite different. I need to understand in the way my book dictates. This proof uses the same concepts but has some ones about chain complexes, which doesn't appear in my book, so it's not exactly the same proof. At 'lemma 20' on the book, it's similar to the $K(dw) + d(Kw) = i_1^*(w)-i_0(w)$ part I need to prove.
UPDATE:
The proof on the PDF and the comments of the book seem very similar. In the book it says to define a function $i_t(x) = (x,t)$ which in the PDF is the function $G_t$. In the book it says for me to define $(Kw)(x) = \int_0^1 \alpha(x,t)\ dt$ while in the PDF it is $\int_0^1 l\frac{\partial }{\partial t} G_i^*(w)$. The book tells me to calculate $K(dw) + d(Kw)$ which is in the PDF the sum of the two integrals. I'm having problem translating from the PDF language to the book language, but I think the proof is exactly the same. Could somebody help me? The greatest difference is the part in the book that says $w = dt\wedge \alpha + \beta$, where $\alpha(x,t) = \sum_{I}\alpha_I(x,t)d_{x_I}$ and $\beta(x,t) = \sum_J b_J(x,t)d_{x_j}$