---- > [!definition] Definition. ([[compactly supported cohomology]]) > Let $X$ be a [[topological space]]; $S \subset X$ a [[subspace topology|subspace]]. Say a ([[singular cohomology|singular]]) [[chain complex of modules|cochain]] $\varphi \in C^{n}(X)$ **has support in $S \subset X$** if > $(\sigma \in C_{n}(X)\text{ restricts to a map } \Delta^{n} \to X - S) \implies \varphi(\sigma)=0$ > i.e., if $\varphi$ kills any [[singular simplex]] $\sigma$ that fails to intersect $S$. > Let $C_{c}^{\bullet} \subset C^{\bullet}(X)$ be the sub-chain complex consisting of those $\varphi$ which have support in *some* [[compact]] $K \subset X$. Call these the **compactly supported cochains in $X$**. Note that[^1] $C_{c}^{\bullet}(X)=\bigcup_{K \subset X \text{ compact}}C^{\bullet}(X, X - K) \subset C^{\bullet}(X).$ Now define the **compactly supported cohomology of $X$** $H_{c}^{*}(X):=H^{*}\big( C_{c}^{\bullet}(X)\big).$ > Compactly supported cohomology is "built out of" those [[relative singular homology|relative cohomology]], not as a union (syntax error), but as a [[categorical colimit|colimit]]:[^2][^3] $H^{n}_{c}(X)=\underset{K \subset X\ \text{compact}}{\operatorname{colim}} H^{n}(X, X - K)$ > In practice, one employs [[computing colimits with cofinals]] to compute $H^{*}_{c}(X)$. See e.g. Example 2 below. > [!basicnonexample] Warning. > $H^{*}_{c}(-)$ is *not* [[homotopy]] invariant! E.g. in Example 2 below it knows about the dimension of $\mathbb{R}^{d}$, since the notion of [[compact|compactness]] is not homotopy invariant. > > Even worse, in general, a map $f:X \to Y$ does not induce a map $f^{*}:H_{c}^{*}(Y) \to H_{c}^{*}(X)$. Indeed, the [[singular (co)chain map and homomorphism induced by a continuous map|usual map]] does not work because preimages need not preserve compactness in general. If we require $f$ to be [[proper map|proper]], [[compactly supported cochain map and morphism on homology induced by a proper map|things work out]]. It turns out that [[inclusion map|inclusions]] of open sets push forward as well: ^warning > [!basicexample] > 1. If $X$ is [[compact]], then $C^{\bullet}_{c}(X)=C^{\bullet}_{c}(X, X - X)=C^{\bullet}(X)$, so $H^{*}_{c}(X)=H^{*}(X)$ is just the usual [[singular cohomology]] of $X$. > 2. We [[(co)homology with coefficients|have]] ($R$ is a, say, [[commutative ring|commutative]] [[ring]]) $H^{i}_{c}\big( \mathbb{R}^{d}; R \big) \cong \begin{cases} R & i = d \\ 0 & \text{otherwise}. \end{cases}$ > >[!proof]- > > > > > > Let $\mathcal{K}(\mathbb{R}^{d})$ be the [[filtered poset]] of compact subsets of $\mathbb{R}^{d}$. Let $\mathcal{B} \subset \mathcal{K}(\mathbb{R}^{d})$ be the balls, namely $\mathcal{B}=\{ N\mathbb{D}^{d} :N =0,1,2,\dots \}$ > > where $N\mathbb{D}^{d}=\overline{B_{N}(0)}$ represents the closed disc in $\mathbb{R}^{d}$ of radius $N$. > > It follows e.g. from [[Heine-Borel theorem|Heine-Borel]] that $\mathcal{B}$ is a [[cofinal monotone map|cofinal subset]] of $\mathcal{K}(\mathbb{R}^{d})$. [[computing colimits with cofinals|Hence]] $H^{n}_{c}(\mathbb{R}^{d}; R)=\underset{K \in \mathcal{K}(\mathbb{R}^{d})}{\operatorname{colim}} H^{n}(\mathbb{R}^{d}, \mathbb{R}^{d} - K; R) \cong \underset{N \in \mathbb{N} \cup \{ 0 \}}{\operatorname{colim}} H^{n}(\mathbb{R}^{d}, \mathbb{R}^{d} - N\mathbb{D}^{d}; R).$ > > We can compute that directly. The [[long exact sequence for relative singular homology|LES]] connecting morphisms $\partial^{*}$ are [[isomorphism|isomorphisms]] in the following diagram (supressing $R$ from the notation): > > > > > > > > ```tikz > > \usepackage{tikz-cd} > > \usepackage{amsmath} > > \usepackage{amsfonts} > > \begin{document} > > % https://tikzcd.yichuanshen.de/#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-UANjtiEpIt2InwOHoVoDeqQAHZQ-yI+woxAY1bZfLFSqoa4fZrtdb9YbBcKhpGhHAY8TSxSU0700IIGgYMSJBBiWB6KIYLwoHxMXwgA > > \begin{tikzcd} > > H^i(\mathbb{R}^d)=0 & H^i(\mathbb{R}^d)=0 \\ > > {H^{i}(\mathbb{R}^d, \mathbb{R}^d - N\mathbb{D}^d)} \arrow[u, "q^*"] \arrow[r, "\rho_{n , n+1 > > }"] & {H^{i}(\mathbb{R}^d, \mathbb{R}^d - (N+1)\mathbb{D}^d)} \arrow[u, "q^*"'] \\ > > H^{i-1}(\mathbb{R}^d - N\mathbb{D}^d) \arrow[u, "\partial^*"] \arrow[r, "\operatorname{id}"] & H^{i-1}(\mathbb{R}^d - (N+1)\mathbb{D}^d) \arrow[u, "\partial^*"'] \\ > > H^{i-1}(\mathbb{R}^d)=0 \arrow[u, "\iota^*"] & H^{i-1}(\mathbb{R}^d)=0 \arrow[u, "\iota^*"'] > > \end{tikzcd} > > \end{document} > > ``` > > > > where the [[singular (co)chain map and homomorphism induced by a continuous map|morphism]] induced by the [[inclusion map|inclusion]] $\mathbb{R}^{d}-(N+1)\mathbb{D}^{d} \hookrightarrow \mathbb{R}^{d}-N\mathbb{D}^{d}$, $H^{i-1}(\mathbb{R}^{d}-N\mathbb{D}^{d}) \to H^{i-1}(\mathbb{R}^{d}-(N+1)\mathbb{D}^{d}),$ > > is the [[identity map]] $\mathbb{Z} \to \mathbb{Z}$ by [[homotopy invariance of singular homology|homotopy invariance]]. Hence, the maps $\rho_{i,i+1}$ comprising our [[diagram|directed system]] indexed by $(\mathbb{N} \cup \{ 0 \} , \leq)$ are all [[isomorphism|isomorphisms]]. The [[categorical colimit|colimit]] of a bunch of isomorphic things is isomorphic to each of them. So $H^{n}_{c}(\mathbb{R}^{d}; R) \cong \underset{N \in \mathbb{N} \cup \{ 0 \}}{\operatorname{colim}} H^{n}(\mathbb{R}^{d}, \mathbb{R}^{d} - N\mathbb{D}^{d}; R) \cong H^{n}(\mathbb{R}^{d} , \mathbb{R}^{d} - \{ 0 \}; R) .$ > > The latter object is just the [[local homology of a manifold|local cohomology]] of $\mathbb{R}^{d}$ at the origin. ---- #### [^1]: When 'algebra brain is turned on' this should be decently clear, but if not, take the following consideration as a hint: let $A \subset X$ and consider the [[group]] $C^n(X,A)=\text{Hom}\big(C_{n}(X,A), \mathbb{Z}\big)=\text{Hom}\big( \frac{C_{n}(X)}{C_{n}(A)}, \mathbb{Z} \big)$. So $\overline{\varphi} \in C^{n}(X,A)$ is a map $\frac{C_{n}(X)}{C_{n}(A)} \to \mathbb{Z}$. By the [[characterization of quotienting a group|universal property of quotient groups]], $\overline{\varphi}$ 'amounts to' a map $\varphi:C_{n}(X) \to \mathbb{Z}$ whose [[kernel of a group homomorphism|kernel]] contains $C_{n}(A)$. ($\varphi(\sigma):=\overline{\varphi}([\sigma])$). This means $\varphi(\sigma)=0$ whenever $\sigma \in C_{n}(A)$, i.e., $\varphi(\sigma)=0$ whenever $\sigma:\Delta^{n} \to A$. In particular, if $A=X-K$, then we see that the elements of $C^{n}(X, X-K)$ are precisely those $\varphi$ for which $\varphi(\sigma)=0$ whenever $\sigma:\Delta^{n} \to X \setminus K$, i.e., those $\varphi$ with support in $K$. (We are using [[dual of quotient is annihilator]] here.) [^2]: Indeed, $C^{\bullet}_{c}(X)=\underset{K \subset X\ \text{compact}}{\operatorname{colim}} C^{\bullet}(X, X - K)$ by definition, and the [[homomorphism on cohomology induced by a cochain map|cohomology functor]] $H^{*}(-)$ [[cohomology functor commutes with colimits|commutes with colimits]]. [^3]: Explicitly, we are taking a [[categorical colimit|colimit]] of a [[diagram|directed system]] over the [[filtered poset]] $\big(\mathcal{K}(X), \subset \big)$. The [[diagram|directed system]] is, of course, the *[[covariant functor|covariant]]* [[covariant functor|functor]] assigning to $K \in \mathcal{K}(X)$ the object $H^{*}(X, X-K)$ and to an [[inclusion map|inclusion]] $K \subset L$ the morphism $H^{*}(X, X-K) \to H^{*}(X, X-L)$ on [[relative singular homology|relative]] [[singular cohomology|cohomology]] [[relative singular homology|induced by]] the [[topological pair|map of pairs]] $(X, X-L) \hookrightarrow (X, X-K)$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` Is the idea that, in general, an inclusion $N_{1} \subset N_{2}$ induces a map $G / N_{1} \to G / N_{2}$ via $gN_{1} \mapsto (gN_{1})N_{2}=g(N_{1}N_{2})=gN_{2}$?