HairyBallTheorem 2002-12-04 02:29:15 2006-09-15 16:50:24 Theorem Poincar\'e-Hopf index theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem*{theorem*}{Theorem} \begin{theorem*} If $X$ is a vector field on $S^{2n}$, then $X$ has a zero. Alternatively, there are no continuous unit vector field on the sphere. Moreover, the tangent bundle of the sphere is nontrivial as a bundle, that is, it is not simply a product. \end{theorem*} There are two proofs for this. The first proof is based on the fact that the antipodal map on $S^{2n}$ is not homotopic to the identity map. The second proof gives the \PMlinkescapetext{hairy ball theorem} as a corollary of the Poincar\'e-Hopf index theorem. Near a zero of a vector field, we can consider a small sphere around the zero, and restrict the vector field to that. By normalizing, we get a map from the sphere to itself. We define the index of the vector field at a zero to be the degree of that map. \begin{theorem*}[Poincar\'e-Hopf index theorem] If $X$ is a vector field on a compact manifold $M$ with isolated zeroes, then $\chi(M)=\sum_{v\in Z(X)}\iota(v)$ where $Z(X)$ is the set of zeroes of $X$, and $\iota(v)$ is the index of $x$ at $v$, and $\chi(M)$ is the Euler characteristic of $M$. \end{theorem*} It is not difficult to show that $S^{2n+1}$ has non-vanishing vector fields for all $n$. A much harder result of Adams shows that the tangent bundle of $S^m$ is trivial if and only if $n=0,1,3,7$, corresponding to the unit spheres in the 4 real division algebras. \begin{proof} First, the low tech proof. Assume that $S^{2n}$ has a unit vector field $X$. Then the \PMlinkname{antipodal map is homotopic to the identity}{AntipodalMapOnSnIsHomotopicToTheIdentityIfAndOnlyIfNIsOdd}. But this cannot be, since the degree of the antipodal map is $-1$ and the degree of the identity map is $+1$. We therefore reject the assumption that $X$ is a unit vector field. This also implies that the tangent bundle of $S^{2n}$ is non-trivial, since any trivial bundle has a non-zero section. \end{proof} \begin{proof} Now for the sledgehammer proof. Suppose $X$ is a nonvanishing vector field on $S^{2n}$. Then by the Poincar\'e-Hopf index theorem, the Euler characteristic of $S^{2n}$ is $\chi(X)=\sum_{v\in X^{-1}(0)}\iota(v)=0$. But the Euler characteristic of $S^{2k}$ is $2$. Hence $X$ must have a zero. \end{proof}