Isolated zero and local index of a vector field

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the canonical smooth tangent-bundle structure.

Let $M$ be a smooth $n$-manifold without boundary, $n\ge 1$, and let $X$ be a smooth vector field on $M$ (A smooth vector field is a smooth section of the tangent bundle, Smooth manifolds and their smooth charts). A point $p\in M$ is an isolated zero of $X$ when $X(p)=0$ and some chart around $p$ contains no other zero of $X$; equivalently, the set of zeros of $X$ has $p$ as an isolated point in a chart around $p$ (Manifold charts, coordinate domains, and coordinate functions).

For an isolated zero choose a smooth chart $(\phi ,U)$ of the smooth structure of $M$ with $\phi (p)=0$ and write $X_{\phi }(u):=(d{\phi }_{{\phi }^{-1}(u)})X({\phi }^{-1}(u))\in {\mathbb{R}}^n$ for the chart representative of $X$ (The induced tangent bundle chart). Since $p$ is an isolated zero there is $\epsilon >0$ with $X_{\phi }\mathrm{\ne }0$ on ${\bar{B}}_{\epsilon }(0)\setminus \{0\}\subseteq \phi (U)$, so the normalized field $v\mapsto X_{\phi }(\epsilon v)\mathrm{/}\mathrm{\mid }{X}_{\phi }(\epsilon v)\mathrm{\mid }$ is a continuous map $S^{n-1}\to S^{n-1}$. The local index of $X$ at $p$ is ${\mathrm{ind}}_{p}X:=\mathrm{deg}(S^{n-1}\to S^{n-1},v\mapsto \frac{X_{\phi }(\epsilon v)}{\mathrm{\mid }{X}_{\phi }(\epsilon v)\mathrm{\mid }}),$ the degree of Degree of a map between oriented closed manifolds computed with the standard orientations of the two copies of $S^{n-1}$, for $n\ge 2$.

For $n=1$ the same formula is read in dimension zero: the sphere $S^0=\{\pm 1\}$ parametrizes $\mathrm{\partial }[-\epsilon ,\epsilon ]$ by $v\mapsto \epsilon v$, and the displayed map is a map $S^0\to S^0$ whose reduced degree in the sense of The reduced degree of a map into the 0-sphere is declared to be the index, ${\mathrm{ind}}_{p}X=\frac{f(+1)-f(-1)}{2}\in \{-1,0,+1\},f(v)=\frac{X_{\phi }(\epsilon v)}{\mathrm{\mid }{X}_{\phi }(\epsilon v)\mathrm{\mid }}\mathrm{.}$ This case is well posed for every $0<{\epsilon }^{\mathrm{\prime }}<\epsilon$: the two signs of $X_{\phi }$ on $(0,\epsilon )$ and on $(-\epsilon ,0)$ are each constant, because $X_{\phi }$ is continuous and nowhere zero on either half-interval, so the two values $f(\pm 1)$ do not depend on ${\epsilon }^{\mathrm{\prime }}$; for $n\ge 2$ the value is independent of $\epsilon$ by the homotopy $v\mapsto X_{\phi }(((1-t)\epsilon +t{\epsilon }^{\mathrm{\prime }})v)\mathrm{/}\mathrm{\mid }\cdot \mathrm{\mid }$ on the zero-free annulus (Degree is invariant under proper smooth homotopy). The value is also independent of the smooth chart and admissible radius, and of trivializations whose fibre orientation matches the chosen base orientation (The local index is independent of chart, ball and trivialization ↗); in particular no orientation of $M$ is required, because a chart change multiplies both the source and the target orientation by the same sign, and the $n=1$ case is the same statement read on the $0$-sphere. Every statement on this page assumes $n\ge 1$; the index is a signed integer, ${\mathrm{ind}}_{p}X\in \mathbb{Z}$.