---- > [!definition] Definition. ([[extension by zero in compactly supported cohomology]]) > Let $X$ be a [[topological space]] and $U \xhookrightarrow{i} X$ an open subset. This induces a map $i_{*}: H^{*}_{c}(U) \to H_{c}^{*}(X)$ on [[compactly supported cohomology]], called **extension by zero**. Explicitly... ^definition Suppose $K \subset U$ is [[compact]]. By [[the excision theorem|excision]], the [[inclusion map|inclusion]] [[topological pair|map of pairs]] $(K, U) \hookrightarrow (K,X)$ [[homomorphism on cohomology induced by a cochain map|induces]] an [[isomorphism]] on [[singular cohomology]] $H^{*}(U, U- K) \xleftarrow[\text{excision}]{\cong} H^{*}(X, X-K).$ Then $i_{*}$ is defined via ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAAkA9AKgH0BjABQBVAJQBeADqSmYWACc4MHMADSAAmlYwmyQFs6OABb9GagL4jR54NIhoY8wxHlg6emMH4QGWPefN1LmAwS2FSdWEAWlVREHNSdExcfEIUAEZyKlpGFjZpWQUlFQ0tHWkDY1MGC0EADWtbSXtHZ1d3T29ff0Dg0PqIupiJLj4hBvjEkAxsPAIiMnTs+mZWRBACuUditV1tXUqTM1UwxrsHJxwXNw8vHz8AoM4QyzrB4fjsmCgAc3giUAAM3kED0SDIICuSAATNQjDA6FA2JAwKxqAw6AAjGAMAAKyTmaRA8iwPyMOBA1BWeXW0m8YB+EWkOBgAA8VGz+FhsARzJMgSCwYhYZCIEh0uYKOYgA \begin{tikzcd} {H^*_c(U)=\underset{K \in \mathcal{K}(U)}{\operatorname{colim}} H^{n}(U, U-K)} \arrow[d, "{\cong, \text{excision}}"', no head] & {\underset{K \in \mathcal{K}(X)}{\operatorname{colim}} H^{n}(X, X-K)=H^*_c(X)} \\ {\underset{K \in \mathcal{K}(U)}{\operatorname{colim}} H^{n}(X, X-K)} \arrow[ru] & \end{tikzcd} \end{document} ``` > [!note] Note. > Note that the definition of $i_{*}$ will always be a bit inexplicit because it involves [[the excision theorem|excision]], which in turn involves the [[small simplices theorem]]. ^note > [!basicexample] > If $U \subset \mathbb{R}^{n}$ is an [[open ball]], then the extension by zero is an [[isomorphism]] $i_{*}:H_{c}^{*}(U) \xrightarrow{\cong}H_{c}^{*}(\mathbb{R}^{n})$. ^basic-example ---- #### - [ ] (this is a more general concept, cf. question on scheme theory exam) ---- #### 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 > ```