---- > [!definition] Definition. ([[singular (co)chain map and homomorphism induced by a continuous map|singular chain map and homomorphism induced by a continuous map]]) > Let $f:X \to Y$ be a [[continuous]] map between [[topological space|topological spaces]]. Then the formula $\begin{align} f_{n}:C_{n}(X)& \to C_{n}(Y) \\ (\sigma : \Delta^{n} \to X) & \mapsto (f \circ \sigma: \Delta^{n} \to Y) \\ \end{align}$ defines a [[chain map]] $f_{\bullet}$ between the [[singular homology|singular chain]] [[chain complex of modules|complexes]] $C_{\bullet}(X)$ and $C_{\bullet}(Y)$, called the **singular chain map induced by $f$**, which [[homomorphism on homology induced by a chain map|in turn]] induces a homomorphism $f_{*}:H_{n}(X) \to H_{n}(Y)$ on [[(co)homology of a complex|homology]] (via the usual [[homomorphism on homology induced by a chain map|homology functor]]). > Compare to [[chain map and homomorphism induced by a simplicial map]]. > > ^definition > [!definition] Definition. ([[singular (co)chain map and homomorphism induced by a continuous map|singular cochain map and homomorphism induced by a continuous map]]) > Let $f:X \to Y$ be a [[continuous]] map between [[topological space|topological spaces]]. Then [[dual map|dualizing]] [[dual vector space|the above]] construction gives a [[chain map|cochain map]] $f^{\bullet}=f_{\bullet}^{\vee}$ between the [[singular cohomology|singular cochain complexes]] $C^{\bullet}(Y)$ and $C^{\bullet}(X)$, called the **singular cochain map induced by $f$**. Explicitly, one has > >$\begin{align}f^{n}:C^{n}(Y) &\to C^{n}(X) \\ \varphi & \mapsto \varphi \circ f_{n}, \text{ i.e.,} \\ f^{n}(\varphi)(\sigma:\Delta^{n} \to X &) = \varphi \circ f \circ \sigma. \end{align}$ This [[homomorphism on cohomology induced by a cochain map|in turn]] induces a homomorphism $f^{*}:H^{n}(Y) \to H^{n}(X)$ on [[(co)homology of a complex|cohomology]] (via the usual [[homomorphism on cohomology induced by a cochain map|cohomology functor]]). > [!justification] > This indeed gives a [[chain map]]. ---- #### ---- #### 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 > ```