记录一下 Poincaré 引理证明的想法, 尤其是链同伦的构造. 我 follow 的是 Bott-Tu 的书 Differential Forms in Algebraic Topology (GTM82). 目前只写了紧支上同调的 Poincaré 引理: \(H_c^{*+1}(M\times \mathbb{R}^1)\cong H_c^*(M)\).
Poincaré Lemma for Compactly Supported Cohomology
Let \(M\) be a smooth manifold. Then the product manifold \(M\times \mathbb{R}^1\) is a smooth manifold as well, and the projection \(\pi:M\times \mathbb{R}^1\to M\) is a smooth map. The pullback \(\pi^*:\Omega^*(M)\to \Omega^*(M\times \mathbb{R}^1)\) does not map \(\Omega_c^*(M)\) into \(\Omega_c^*(M\times \mathbb{R}^1)\), so instead we consider the pushforward known as integration along the fiber
\[\begin{equation*}
\pi_*:\Omega_c^*(M\times \mathbb{R}^1)\to \Omega_c^{*-1}(M)\quad \begin{cases}
f(x,t)\pi^*\phi\mapsto 0\\
f(x,t)\mathrm{d}{t}\wedge\pi^*\phi\mapsto \left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi
\end{cases}
\end{equation*}
\]
where \(\phi\in \Omega^*(M)\) and \(f\in C_c^{\infty}(M\times \mathbb{R};\mathbb{R})\). (These two types of elements in \(\Omega_c^*(M\times \mathbb{R}^1)\) will soon be revisited.) It's quick to check that \(\pi_*\) is a chain map. We will show that the induced map \(\pi_*:H_c^*(M\times \mathbb{R}^1)\to H_c^{*-1}(M)\) is an isomorphism.
Take \(e=e(t)\mathrm{d}{t}\in \Omega_c^1(\mathbb{R}^1)\) such that \(\displaystyle\int_{\mathbb{R}^1} e=1\). Define
\[\begin{equation*}
e_*:\Omega_c^*(M)\to \Omega_c^{*+1}(M\times \mathbb{R}^1)\quad \psi\mapsto e(t)\mathrm{d}{t}\wedge\pi^*\psi
\end{equation*}
\]
where the map \(t\mapsto e(t)\) has been identified with the composition \((x,t)\mapsto t\mapsto e(t)\). It's quick to check that \(e_*\) is a chain map, and \(\pi_*\circ e_*=\text{id}\) on \(\Omega_c^*(M)\). We will show that there is a chain homotopy \(K\) connecting \(e_*\circ \pi_*\) and \(\text{id}\) on \(\Omega_c^*(M\times \mathbb{R}^1)\), so that the induced map \(e_*:H_c^{*}(M)\to H_c^{*+1}(M\times \mathbb{R}^1)\) is the inverse of \(\pi_*:H_c^*(M\times \mathbb{R}^1)\to H_c^{*-1}(M)\).
We shall construct the desired chain homotopy \(K:\Omega_c^*(M\times \mathbb{R}^1)\to \Omega_c^{*-1}(M\times \mathbb{R}^1)\) from the basic relation
\[\begin{equation*}
\text{id}-e_*\circ \pi_*=\mathrm{d}K+K \mathrm{d}
\end{equation*}
\]
To proceed, interpret this relation on the aforementioned two types of elements in \(\Omega_c^*(M\times \mathbb{R}^1)\), and be reminded that computations will be done w.r.t. some specific local coordinates.
- For \(f(x,t)\pi^*\phi\), we require
\[ \begin{equation*}
f(x,t)\pi^*\phi=\mathrm{d}K(f(x,t)\pi^*\phi)+K \left[\left(\displaystyle\sum_i \dfrac{\partial f}{\partial x^i}(x,t)\mathrm{d}{x^i}+\dfrac{\partial f}{\partial t}(x,t)\mathrm{d}{t}\right)\wedge \pi^*\phi+f(x,t)\pi^*\mathrm{d}{\phi}\right]
\end{equation*}
\]
- For \(f(x,t)\mathrm{d}{t}\wedge\pi^*\phi\), we require
\[ \begin{align*}
&f(x,t)\mathrm{d}{t}\wedge\pi^*\phi-e(t)\mathrm{d}{t}\wedge\pi^*\left[\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi\right]\\
&=\mathrm{d}K(f(x,t)\mathrm{d}{t}\wedge\pi^*\phi)+K \left[-\displaystyle\sum_i \dfrac{\partial f}{\partial x^i}(x,t)\mathrm{d}{t}\wedge\mathrm{d}{x^i}\wedge\pi^*\phi-f(x,t)\mathrm{d}{t}\wedge\pi^*\mathrm{d}{\phi}\right]
\end{align*}
\]
We observe that when \(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t=0\), or equivalently, \(f\in \dfrac{\partial }{\partial t}(C_c^{\infty}(M\times \mathbb{R}^1))\), the two requirements become very close. An immediate guess follows:
\[\begin{equation*}
K_1(f(x,t)\pi^*\phi)=0,\quad K_1(f(x,t)\mathrm{d}{t}\wedge\pi^*\phi)=\left(\displaystyle\int_{-\infty}^{t} f(x,t) \,\mathrm{d}t\right)\pi^*\phi
\end{equation*}
\]
whenever \(\phi\in \Omega^*(M)\) and \(f\in C_c^*(M\times \mathbb{R}^1)\). While \(K_1\) solves the first requirement, it does not reconcile with the second:
\[\begin{align*}
&\mathrm{d}K_1(f(x,t)\mathrm{d}{t}\wedge\pi^*\phi)+K_1 \left[-\displaystyle\sum_i \dfrac{\partial f}{\partial x^i}(x,t)\mathrm{d}{t}\wedge\mathrm{d}{x^i}\wedge\pi^*\phi-f(x,t)\mathrm{d}{t}\wedge\pi^*\mathrm{d}{\phi}\right]\\
=&\mathrm{d}\left[\left(\displaystyle\int_{-\infty}^{t} f(x,t) \,\mathrm{d}t\right)\pi^*\phi\right]-\displaystyle\sum_i \left(\displaystyle\int_{-\infty}^{t} \dfrac{\partial f}{\partial x^i}(x,t) \,\mathrm{d}t\right)\mathrm{d}{x^i}\wedge \pi^*\phi-\left(\displaystyle\int_{-\infty}^{t} f(x,t) \,\mathrm{d}t\right)\pi^*\mathrm{d}{\phi}\\
=&f(x,t)\mathrm{d}{t}\wedge \pi^*\phi
\end{align*}
\]
However, the computation also indicates that we are not far away from success, with only one term involving \(e=e(t)\mathrm{d}{t}\) missing.
As a remedy, consider \(K_2=K-K_1\). It suffices to construct \(K_2\) in the same fashion:
- For \(f(x,t)\pi^*\phi\), we require
\[ \begin{equation*}
0=\mathrm{d}K_2(f(x,t)\pi^*\phi)+K_2 \left[\left(\displaystyle\sum_i \dfrac{\partial f}{\partial x^i}(x,t)\mathrm{d}{x^i}+\dfrac{\partial f}{\partial t}(x,t)\mathrm{d}{t}\right)\wedge \pi^*\phi+f(x,t)\pi^*\mathrm{d}{\phi}\right]
\end{equation*}
\]
- For \(f(x,t)\mathrm{d}{t}\wedge\pi^*\phi\), we require
\[ \begin{align*}
&-e(t)\mathrm{d}{t}\wedge\pi^*\left[\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi\right]\\
&=\mathrm{d}K_2(f(x,t)\mathrm{d}{t}\wedge\pi^*\phi)+K_2 \left[-\displaystyle\sum_i \dfrac{\partial f}{\partial x^i}(x,t)\mathrm{d}{t}\wedge\mathrm{d}{x^i}\wedge\pi^*\phi-f(x,t)\mathrm{d}{t}\wedge\pi^*\mathrm{d}{\phi}\right]
\end{align*}
\]
It is again natural to set \(K_2(f(x,t)\pi^*\phi)=0\), and so the first requirement simplifies to
\[\begin{equation*}
K_2 \left(\dfrac{\partial f}{\partial t}(x,t)\mathrm{d}{t}\wedge \pi^*\phi\right)=0
\end{equation*}
\]
Now we focus on the second requirement. Note that
\[\begin{align*}
&\hphantom{=}e(t)\mathrm{d}{t}\wedge\pi^*\left[\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi\right]\\
&=\mathrm{d}\left(\displaystyle\int_{-\infty}^{t} e(t) \,\mathrm{d}t\right)\wedge\pi^*\left[\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi\right]+\left(\displaystyle\int_{-\infty}^{t} e(t) \,\mathrm{d}t\right)\mathrm{d}\pi^*\left[\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi\right]\\
&\hphantom{=\mathrm{d}\left(\displaystyle\int_{-\infty}^{t} e(t) \,\mathrm{d}t\right)\wedge\pi^*\left[\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi\right]}-\left(\displaystyle\int_{-\infty}^{t} e(t) \,\mathrm{d}t\right)\mathrm{d}\pi^*\left[\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi\right]\\
&=\mathrm{d}\left\{\left(\displaystyle\int_{-\infty}^{t} e(t) \,\mathrm{d}t\right)\pi^*\left[\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi\right]\right\}\\
&\hphantom{=}-\left(\displaystyle\int_{-\infty}^{t} e(t) \,\mathrm{d}t\right)\pi^*\left[\displaystyle\sum_i \left(\displaystyle\int_{-\infty}^{\infty} \dfrac{\partial f}{\partial x^i}(x,t) \,\mathrm{d}t\right)\mathrm{d}{x^i}\wedge \phi+\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\mathrm{d}{\phi}\right]
\end{align*}
\]
An immediate guess follows:
\[\begin{equation*}
K_2(f(x,t)\mathrm{d}{t}\wedge \pi^*\phi)=-\left(\displaystyle\int_{-\infty}^{t} e(t) \,\mathrm{d}t\right)\pi^*\left[\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi\right]
\end{equation*}
\]
which indeed solves the second requirement, and together with \(K_2(f(x,t)\pi^*\phi)=0\), solves the first, since
\[\begin{equation*}
\displaystyle\int_{-\infty}^{\infty} \dfrac{\partial f}{\partial t}(x,t) \,\mathrm{d}t=0
\end{equation*}
\]
Conclusion: The map \(K: \Omega_c^{*}(M\times \mathbb{R}^1)\to \Omega_c^{*-1}(M\times \mathbb{R}^1)\) defined by
\[\begin{equation*}
\begin{cases}
f(x,t)\pi^*\phi\mapsto 0\\
f(x,t)\mathrm{d}{t}\wedge\pi^*\phi\mapsto \left(\displaystyle\int_{-\infty}^{t} f(x,t) \,\mathrm{d}t\right)\pi^*\phi-\left(\displaystyle\int_{-\infty}^{t} e(t) \,\mathrm{d}t\right)\pi^*\left[\left(\displaystyle\int_{-\infty}^{\infty} f(x,t) \,\mathrm{d}t\right)\phi\right]
\end{cases}
\end{equation*}
\]
is a chain homotopy connecting \(e_*\circ \pi_*\) and \(\text{id}\) on \(\Omega_c^*(M\times \mathbb{R}^1)\), and consequently the maps
\[\begin{equation*}
H_c^{*+1}(M\times \mathbb{R})\begin{matrix}\xrightarrow{\pi_*}\\ \xleftarrow[e_*]{}\end{matrix}H_c^*(M)
\end{equation*}
\]
are isomorphisms.
As a corollary, for each positive integer \(n\), there holds
\[\begin{equation*}
H_c^*(\mathbb{R}^n)=\begin{cases}
\mathbb{R}, &\text{in dimension $n$}\\
0, &\text{elsewhere}
\end{cases}
\end{equation*}
\]
where the isomorphism \(H_c^*(\mathbb{R}^n)\xrightarrow{\cong}\mathbb{R}\) is given by iterated \(\pi_*\), i.e., by integration over \(\mathbb{R}^n\), thanks to Fubini's theorem. Besides, by iterating \(e_*\), we see that a generator of \(H_c^n(\mathbb{R}^n)\) is represented by a bump \(n\)-form \(\alpha=\alpha(x)\mathrm{d}{x^1}\wedge\cdots\wedge \mathrm{d}{x^n}\) with \(\displaystyle\int_{\mathbb{R}^n} \alpha=1\), whose support can be made as small as possible.
