---- > [!theorem] Theorem. ([[hairy ball theorem]]) > The $n$-sphere $\mathbb{S}^{n}$ has a nowhere-vanishing [[vector field]] if and only if $n=2k+1$ is odd. > ![[Pasted image 20250512180042.png]] ^theorem > [!note] Remark. > Since [[the Euler characteristic of an odd-dimensional compact manifold is zero]], this implies that if $\mathbb{S}^{n}$ has a nowhere-vanishing [[vector field]], then its [[Euler characteristic of a cell complex|Euler characteristic]] $\chi(\mathbb{S}^{n})=0$. This in fact holds for all [[smooth manifold|smooth]] [[compact]] [[orientable manifold|oriented]] [[manifold|manifolds]], as can be seen by using that[^1] $\chi(M)=e(TM) [M]$ alongside [[Euler class obstructs existence of nonvanishing section]]. ^note [[degree of a continuous map on the sphere|degree]] > [!proof]- Proof. ([[hairy ball theorem]]) > Suppose $X \in \Gamma(TM)$ is never zero. Then we are safe to normalize: set $Y(x)= \frac{X(x)}{\|X(x)\|}$, (Viewing $T_{x}M \subset \mathbb{R}^{n+1}$) yielding a [[vector field]] which may be viewed as a map $\mathbb{S}^{n} \to \mathbb{S}^{n}$. > > We can now construct a [[homotopy]] from $Y$ to the [[antipodal map]] by "linear interpolation, but staying on the sphere". > > Indeed, let $\begin{align} > H: [0, \pi] \times \mathbb{S}^{n} &\to \mathbb{S}^{n} \\ > (t, x ) &\mapsto \cos(t) x + \sin(t) Y(x) > \end{align}$ > this defines a [[homotopy]] from $\id_{\mathbb{S}^{n}}$ to the [[antipodal map]] $a:\mathbb{S}^{n} \to \mathbb{S}^{n}$.[^2] Indeed, given $x \in \mathbb{S}^{n}$: > - $H(0, x)=1 \cdot x + 0 \cdot Y(x)=x$ > - $H(\pi, x)=-1 \cdot x + 0 \cdot Y(x)=a(x)$ > > 'On the way', $H$ defines a [[homotopy]] from $Y$ to $a$, because > - $H\left( \frac{\pi}{2}, x \right)=0 \cdot x + 1 \cdot Y(x)=Y(x)$. > > > Since [[homotopy equivalent|homotopic maps induce the same map on homology]], we have $1=\text{deg }\id_{\mathbb{S}^{n}}=\text{deg }a= (-1)^{n+1},$ > where we have used [[degree of the antipodal map]], implying $n$ is odd. > > > **Existence when $n=2k+1$ is odd.** When $n=2k+1$ is odd, the following 'rotation of tangents' vector field $X$ is nonvanishing (draw a picture with $\mathbb{S}^{1}$): $X(x_{1},y_{1},x_{2},y_{2},\dots,y_{k}):= (-y_{1}, x_{1}, -y_{2}, x_{2}, \dots, x_{k}).$ > ---- #### Note: DG lecture notes proved a version of this, but not the actual lectures. [^1]: $e(TM)$ is the [[The Thom isomorphism theorem|Euler class]] of the [[tangent bundle]] $TM$ and $[M]$ is the [[The Thom Theorem for oriented manifolds|fundamental class]] of $M$. [^2]: Compute: $\begin{align} \|\cos(t) x + \sin(t) Y(x) \|^{2} &= \langle \cos(t) x + \sin(t) Y(x) , \cos(t) x + \sin(t) Y(x) \rangle \\ &= \cos ^{2}(t) \cancel{ \|x\|^{2} }^{=1} + 2 \cancel{ \langle \sin(t)Y(x), \cos(t) x \rangle } ^{=0, x \perp Y(x)} + \sin ^{2}(t) \cancel{ \|Y(x)\|^{2} }^{=0} \\ &= 1. \end{align}$ ----- #### 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 > ```