---- > [!definition] Definition. ([[local degree]]) (Oscar's version) > Let $f:\mathbb{S}^{d} \to \mathbb{S}^{d}$ be a map, and $x\in \mathbb{S}^{d}$. > The [[topological pair|map of pairs]] $(\mathbb{S}^{d}, \{ x \}) \xrightarrow{f} (\mathbb{S}^{d}, f(x))$ [[relative singular homology|induces a map]] on [[relative singular homology]] $f_{*}:H_{i}\big( \mathbb{S}^{d}, \mathbb{S}^{d}-\{ x \} \big) \to H_{i}\big(\mathbb{S}^{d}, \mathbb{S}^{d}-\{ f(x) \}\big).$ We identify $H_{d}\big( \mathbb{S}^{d}, \mathbb{S}^{d} - \{ x \} \big) \cong H_{d}(\mathbb{S}^{d}) \cong \mathbb{Z}$ for any $x$ [[long exact sequence for relative singular homology|via]] $\cancel{ H_{d}(\{ x \}) }^{=0} \xrightarrow{\iota_{*}} \underbrace{ H_{d}(\mathbb{S}^{d}) }_{ \cong \mathbb{Z} } \xrightarrow{q_{*}, \cong} H_{d}(\mathbb{S}^{d}, \mathbb{S}^{d}-\{ x \}) \xrightarrow{\partial_{}}\cancel{ H_{d-1}(\{ x \}) }^{=0}.$ Then $f_{*}$ is given by multiplication by a constant $\text{deg}(f)_{x}$, called the **local degree of $f$ at $x$**. ^definition > [!definition] Definition. ([[local degree]]) (Jan's version) > Suppose $f:\mathbb{S}^{d} \to \mathbb{S}^{d}$ such that the fiber $f^{-1}(\{ y \})=\{ x_{1},\dots,x_{k} \}$ is finite. Pick disjoint [[neighborhood|neighborhoods]] $U_{i}\ni x_{i}$, $U_{i} \subset \mathbb{S}^{d}$ open, such that each $f(U_{i}) \subset V$ for some open neighborhood $V \ni y$.[^2] > > ![[Pasted image 20250513103106.png]] > > The restriction of $f$ to one of the $U_{i}$, $f |_{U_{i}}$, gives a [[topological pair|map of pairs]] $(U_{i}, U_{i}-\{ x_{i} \}) \to (V, V-\{ y \})$[^1] [[relative homology of an embedding of chain complexes|from which]] we get the following commutative diagram[^3]: > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAAkB9KACgFVOWUgAIBWYQFphAHWnAAHoNkBfEMtLpMufIRQBGclVqMWbLrwBqIi5JlyAnirUaQGbHgJEyeo-WatEDm4eWQBbOhwACwAjaOAAZWUAPSgRMIiYuMSU21kFFQBKZ013HSIDH2o-U0DzEOlwqNiE5NS7RsyWnKk8x2llIvUS7U8UMgAmXxMAkFkmMFgAJ2jFugBjGGA69Kas1oLlTmAAXh3OgC1lZWFi1y0PXWQDSarptjmFmGXVja3gs+a2SgByOpwaGWal2uaiMMCgAHN4ERQAAzRYQUJIMggHAQJAGYz+Ng8FHCAA+RzEA04ACpbmiMVjqLikONXkTArIcDB5DhgFgwGsGIcaWlpGsCPCxdzecAeWssNgCKpqAw6NEYAwAAr3MqBRZYeGRHD09GYxAElmIADM7Jqs2kMr5AqFIrFErAUrsTrl8gVSrAqiGIAZ5oALMy8Yg2YT7QBHWnuyWmxmIACskaQttjMwTorsHvhIFV6s1OtKoxABqNJuDoaQEZxUYzOfe0k4BagEBw3p5fNg8OUJIK5KOiiwQYoyiAA > \begin{tikzcd} > {H_d(U_i, U_i - \{x_i\}} \arrow[r, "(f |_{U_i})_*"] \arrow[d, "{\text{incl}_*, \cong, \text{excision}}"'] & {H_d(V, V - \{y\}} \arrow[d, "{\text{incl}_*, \cong, \text{excision}}"] \\ > {H_d(\mathbb{S}^d, \mathbb{S}^d - \{x\})} & {H_d(\mathbb{S}^d, \mathbb{S}^d - \{y\})} \\ > \underbrace{H_d(\mathbb{S}^d)}_{=\mathbb{Z}} \arrow[u, "{q_*, \cong}"] \arrow[r, "\_ \cdot \text{deg}(f) |_{x_i}"] & \underbrace{H_d(\mathbb{S}^d)}_{=\mathbb{Z}} \arrow[u, "{q_*, \cong}"'] > \end{tikzcd} > \end{document} > ``` > > We define the **local degree of $f$ at $x_{i}$** as $\text{deg}(f) |_{x_{i}}$ in the diagram. It is the integer by which $f_{*}$ multiplies in [[local homology of a manifold|local homology]] at $x_{i} \in f ^{-1}(\{ y \})$. > ---- #### [^1]: We have $f(U-\{ x_{i} \}) \subset f(V - \{ y \})$ because the $U_{i}$ are disjoint (so $y \not \in f(U_{i}-\{ x_{i} \})$). [^2]: Of course, we can choose $Y$ to be all of $\mathbb{S}^{n}$... but it will be later useful to have that $Y$ could be chosen small. [^3]: The $q_{*}$ maps in the upward arrows come from the [[long exact sequence for relative singular homology|long exact sequence for the pair]] $(\mathbb{S}^{d}, \mathbb{S}^{d}-\{ x \})$, $\cancel{ H_{d}(\mathbb{S}^{d}-\{ x \}) }^{=0}\xrightarrow{\iota_{*}} H_{d}(\mathbb{S}^{d}) \xrightarrow{q_{*}} H_{d}(\mathbb{S}^{d}, \mathbb{S}^{d}-\{ x \}) \xrightarrow{\partial_{}} \cancel{ H_{d-1}(\mathbb{S}^{d}-\{ x \}) }^{=0}.$This recovers the first local degree definition. ---- #### 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 > ```