Compactly Supported de Rham Cohomology

Poincaré Lemma with Compact Support: Let $1\le p\le n$, and suppose $\omega$ is a compactly supported closed $p$-form on ${\mathbb{R}}^n$. If $p=n$, suppose in addition that $$\int_{\Bbb R^n}\omega = 0.$$Then there exists a compactly supported smooth $(p-1)$-form $\eta$ on ${\mathbb{R}}^n$ such that $d\eta =\omega$.

Def: Let $M$ be a smooth manifold with or without boundary and let ${\mathrm{\Omega }}_c^p(M)$ denote the vector space of compactly supported smooth $p$-forms on $M$. The $p$th compactly supported de Rham cohomology group of $M$ is the quotient space $$H_c^p(M) := \frac{\ker (d : \Omega^p_c(M) \to \Omega^{p+1}_c(M))}{\text{Im}(d: \Omega^{p-1}_c (M) \to \Omega^p_c(M))}. $$Of course, when $M$ is compact, this just reduces to ordinary de Rham cohomology.

Compactly Supported Cohomology of ${\mathbb{R}}^n$: For $n\ge 1$, the compactly supported de Rham cohomology groups of ${\mathbb{R}}^n$ are $$H_c^p(\Bbb R^n) \cong \begin{dcases}
0 & 0\le p < n, \
\Bbb R & p = n.
\end{dcases}$$
We see that a smooth map doesn't pull back compactly supported forms to compactly supported smooth ones. so it doesn't induce a map on compactly supported cohomology. A proper map does pull back compactly supported forms to compactly supported ones, so for a proper smooth map $F:M\to N$ there is an induced cohomology map $F^{\ast }:H_c^p(N)\to H_c^p(M)$ for each $p$.

Top Cohomology, Orientable Compact Support Case: If $M$ is a connected oriented smooth $n$-manifold, then the integration map $I:H_c^n(M)\to \mathbb{R}$ is an isomorphism, so $H_c^n(M)$ is $1$-dimensional.

Top Cohomology, Orientable Compact Case: If $M$ is a compact connected orientable smooth $n$-manifold, then $H_{\text{dR}}^n(M)$ is $1$-dimensional, and is spanned by the cohomology class of any smooth orientation form.

Top Cohomology, Orientable Noncompact Case: If $M$ is a noncompact connected orientable smooth $n$-manifold, then $H_{\text{dR}}^n(M)=0$.

Lemma: Suppose $M$ is a connected nonorientable smooth manifold and $\hat{\pi }:\hat{M}\to M$ is its orientation covering. For each $p$, the induced cohomology maps ${\hat{\pi }}^{\ast }:H_{\text{dR}}^p(M)\to H_{\text{dR}}^p(\hat{M}),$ and ${\hat{\pi }}^{\ast }:H_c^p(M)\to H_c^p(\hat{M})$ are injective.

Top Cohomology, Nonorientable Case: If $M$ is a connected nonorientable smooth $n$-manifold, then $H_c^n(M)=0$ and $H_{\text{dR}}^n(M)=0$.

Prop: Let $M$ be a connected smooth manifold of dimension $n\ge 3$. For any $x\in M$ and $0\le p\le n-2$, the map $H_{\text{dR}}^p(M)\to H_{\text{dR}}^p(M\setminus \{x\})$ induced by the inclusion $M\setminus \{x\}\hookrightarrow M$ is an isomorphism. If in addition, $M$ is compact and orientable then it is true when $p=n-1$.

Cor: Let $M_1,M_2$ be connected smooth manifolds of dimension $n\ge 3$, and let $M_1\mathrm{\#}{M}_2$ denote their smooth connected sum. Then $H_{\text{dR}}^p(M_1\mathrm{\#}{M}_2)\cong H_{\text{dR}}^p(M_1)\oplus H_{\text{dR}}^p(M_2)$ for $0<p<n-1$. If in addition, $M_1$ and $M_2$ are compact and orientable, then it is also true for $p=n-1$.

Prop: Suppose $M$ is a compact, connected, orientable, smooth $n$-manifolds.

• There is a one-to-one correspondence between orientations of $M$ and orientations of the vector space $H_{\text{dR}}^n(M)$, under which the cohomology class of a smooth orientation form is an oriented basis for $H_{\text{dR}}^n(M)$.
• Suppose $M$ and $N$ are smooth $n$-manifolds with given orientations. A diffeomorphism $F:M\to N$ is orientation preserving iff $F^{\ast }:H_{\text{dR}}^n(N)\to H_{\text{dR}}^n(M)$.

Lemma: Given an open subset $U\subseteq M$, let $\iota :U\hookrightarrow M$ denote the inclusion map, and define a linear map ${\iota }_{\mathrm{♯}}:{\mathrm{\Omega }}_c^p(U)\to {\mathrm{\Omega }}_c^p(M)$ by extending each compactly supported from to be zero on $M\setminus U$. Then we get that $d\circ {\iota }_{\mathrm{♯}}={\iota }_{\mathrm{♯}}\circ d$, and so ${\iota }_{\mathrm{♯}}$ induces a linear map on compactly supported cohomology, denoted by ${\iota }_{\ast }:H_c^p(U)\to H_c^p(M)$.

Mayer-Vietoris with Compact Supports: Suppose $M$ is a smooth manifold and $U,V\subseteq M$ are open subsets whose union is $M$. For each nonnegative integer $p$, there is a linear map ${\delta }_{\ast }:H_c^p(M)\to H_c^{p+1}(U\cap V)$ such that the following sequence is exact $$\cdots
\stackrel{\delta_}{\longrightarrow} H_c^p(U \cap V) \stackrel{i_ \oplus(-j_)}{\longrightarrow}H_c^p(U) \oplus H_c^p(V)\stackrel{k_+ l_}{\longrightarrow} H_c^p(M) \stackrel{\delta_}{\longrightarrow} H_c^p(U \cap V) \stackrel{i_* \oplus(-j_*)}{\longrightarrow}\cdots,$$where $i,j,k,l$ are the inclusion maps

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\begin{document}
\begin{tikzcd}
& U\arrow[dr, "k"] & \\
U \cap V\arrow[ur, "i"]\arrow[dr, "j"] && M \\
&V\arrow[ur,"\ell"']&
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Cor: Let $H_c^p(M{)}^{\ast }$ denote the algebraic dual space of $H_c^p(M)$. The following sequence is also exact:$$\cdots \stackrel{(\delta_)^}{\longrightarrow} H_c^p(M)^* \stackrel{(k_)\oplus (l_)^* }{\longrightarrow} H_c^p(U) \oplus H_c^p(V)^* \stackrel{(i_)^-(j_)^}{\longrightarrow} H_p^c(U \cap V)^* \stackrel{(\delta_)^}{\longrightarrow} H^{p-1}c(M)^*\stackrel{(k)\oplus (l_)^* }{\longrightarrow}\cdots. $$
Def: Let $M$ be an oriented smooth $n$-manifold. We define a map $\text{PD}:{\mathrm{\Omega }}^p(M)\to {\mathrm{\Omega }}_c^{n-p}(\mathrm{\Omega }{)}^{\ast }$ by $$\text{PD}(\omega)(\eta) := \int_M \omega \wedge \eta. $$
The Poincaré Duality Theorem: $\text{PD}$ descends to an isomorphism $\text{PD}:H_{\text{dR}}^p(M)\to H_c^{n-p}(M{)}^{\ast }.$

Prop: Let $M$ be a compact smooth $n$-manifold.

• All of the de Rham groups of $M$ are finite-dimensional.
• If $M$ is orientable, then $\dim H_{\text{dR}}^p(M)=\dim H_{\text{dR}}^{n-p}(M)$ for all $p$.

Cor: If $M$ is a compact, orientable, and odd dimensional smooth manifold, then $\chi (M)=0.$