----- > [!proposition] Proposition. ([[local homology of a manifold]]) > Let $M$ be a [[topological manifold]] of dimension $d$. Then $H_{i} (M, M - \{ x \}) \cong \begin{cases} \mathbb{Z} & i=d \\ 0 & \text{else.} \end{cases}$ Here, $H_{i}(M,M-\{ x \})$ denotes the $i$th [[(co)homology of a complex|homology]] of $M$ [[relative singular homology|relative]] to 'everything but $x. Intuitively, this means we are 'ignoring $M$ away from $x and it is therefore called the **local homology of $M$ at $x$**. Sometimes denoted $H_{i}(M | x)$. > > More generally, if $K \subset X$ then we can speak of the **local homology of $M$ near $K$**, $H_{*}(X | K):= H_{*}(X, X-K).$ > The **local cohomology** groups $H^{*}(X | K)$ are defined in the same way. Note that $H^{*}(X | \{ x \})$ is [[dual vector space|dual]] to $H_{*}(X | \{ x \})$ . > > > > Note that if $K$ is [[closed set|closed]], then [[the excision theorem|excision]] implies that the local (co)cohomology depends only on some neighborhood of $K$, i.e., $K \subset U \implies H^{*}(M | K) \cong H^{*}(U | K)$ > (in the case of local homology at a point $x$, this $U$ is just a coordinate neighborhood — that's how the proof below proceeds.) > [!proof]- Proof. ([[local homology of a manifold]]) > Let $U \ni x$ be a [[coordinate chart|chart]] centered at $x$, $U \cong \mathbb{R}^{d}$. By [[the excision theorem]] (corollary), $H_{i}(M, M-\{ x \}) \cong H_{i}(U, U - \{ x \})$ and per [[homotopy invariance of singular homology|homotopy invariance]] this is isomorphic to $H_{i}(\mathbb{R}^{d}, \mathbb{R}^{d}-\{ 0 \}).$ This we can compute by fitting it into the [[long exact sequence for relative singular homology|long exact sequence for a pair]]: $\overbrace{H_{i}(\mathbb{R}^{d}- \{ 0 \})}^{=\begin{cases} \mathbb{Z} & i = d-1 \\ 0 & \text{else.} \end{cases}} \to \cancel{H_{i}(\mathbb{R}^{d})}^{0} \to H_{i}(\mathbb{R}^{d}, \mathbb{R}^{d}- \{ 0 \}) \to \overbrace{H_{i-1}(\mathbb{R}^{d}- \{ 0 \})}^{=\begin{cases} \mathbb{Z} & i-1 = d-1 \\0 & \text{else.}\end{cases}}$ the only interesting stuff happens around $i=d$, where we get an [[exact sequence]] $0 \to 0 \to H_{d}(\mathbb{R}^{d}, \mathbb{R}^{d}- \{ 0 \} ) \to \mathbb{Z} \to 0$ from which the result follows. (Technically we only showed for $i \geq 2$. We need to check the lower-degree terms manually, but we shall not do so here.) ----- #### [^1]: This is general fact, see [[the excision theorem|excision]]. Argument repeated here. $H_{i}(M, M-\{ x \}) \cong H_{i}(M - Z, (M-\{ x \})-Z)$where $Z=M-U$. But $M-Z=U$, and $(M-\{ x \})-Z=U-\{ x \}$. So this equals $H_{i}(U, U - \{ x \}).$ ---- #### 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 > ```