---- > [!definition] Definition. ([[degree of a map between manifolds]]) > Let $f:M \to N$ be a [[continuous|map]] between [[compact]], ($\mathbb{Z}$-)[[(homological) orientation of a manifold|oriented]], [[connected]] $n$-[[manifold|manifolds]]. Denote by $[M]$, $[N]$ the respective [[The Thom Theorem for oriented manifolds|fundamental classes]]. Then (using [[Poincare duality]]) $f_{*}[M] \in H_{n}(N) \xleftarrow[\text{P.D.}]{\cong} H^{0}(N) \cong \mathbb{Z}\langle [N] \rangle ,$ meaning $f_{*}[M]=k \cdot [N]$ for some integer $k$. We call $k$ the **degree** of $f$. > >By exactly the same proof, we can compute this degree using [[local degree|local degrees]], [[degree is sum of local degrees|just as in the case of]] [[degree of a continuous map on the sphere|the sphere]]. ^definition > [!specialization] > - [ ] explain specialization to [[degree of a continuous map on the sphere]] ^specialization ---- #### ---- #### 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 > ```