Isolated fixed point and local fixed point index

Let $M$ be a smooth $n$-manifold without boundary, $n\ge 1$ (Smooth manifolds and their smooth charts), let $f:M\to M$ be a smooth map ($C^r$ and smooth maps between smooth manifolds) and let $x$ be an isolated fixed point of $f$, i.e. some neighbourhood of $x$ contains no other fixed point. Choose a smooth chart $(\phi ,U)$ from the smooth atlas of $M$ (Smooth manifolds and their smooth charts) with $x\in U$ and $\phi (x)=0$ (Manifold charts, coordinate domains, and coordinate functions) and $\epsilon >0$ such that the closed ball $\overline{B_{\epsilon }(0)}$ lies in $\phi (U\cap f^{-1}(U))$ and $\widehat{f}(u):=\phi (f({\phi }^{-1}(u)))\mathrm{\ne }u$ for $0<\mathrm{\mid }u\mathrm{\mid }\le \epsilon$, and set $g(u):=u-\widehat{f}(u)$. The local fixed point index of $f$ at $x$ is the degree

${\mathrm{ind}}_x(f):=\mathrm{deg}(S^{n-1}\to S^{n-1},v\mapsto \frac{g(\epsilon v)}{\mathrm{\mid }g(\epsilon v)\mathrm{\mid }})\in \mathbb{Z}$

of Degree of a map between oriented closed manifolds for $n\ge 2$, both spheres carrying their boundary orientations. For $n=1$ use the reduced degree of The reduced degree of a map into the 0-sphere: if $h(v)=g(\epsilon v)\mathrm{/}\mathrm{\mid }g(\epsilon v)\mathrm{\mid }$, then ${\mathrm{ind}}_x(f)=(h(+1)-h(-1))\mathrm{/}2$. Radius independence follows by radial interpolation in the zero-free punctured ball, using Degree is invariant under proper smooth homotopy for $n\ge 2$ and Reduced degree into the 0-sphere is homotopy invariant and multiplicative for $n=1$ and of the chart, the ball and the neighbourhood by The local fixed point index is independent of chart, ball and neighbourhood ↗; it uses no orientation of $M$, because a chart change multiplies source and target orientations by the same sign. The empty sum over a fixed-point-free map is $0$ by convention, and this local-index definition is restricted to $n\ge 1$.

• Why a radius can be chosen. Since $x$ is isolated and $\phi$ is a homeomorphism with $\phi (x)=0$, the representative $\widehat{f}$ is defined on the open neighbourhood $\phi (U\cap f^{-1}(U))$ of $0$. Isolation excludes other zeros there. For $\epsilon$ small enough that $\{\mathrm{\mid }u\mathrm{\mid }\le \epsilon \}$ lies in that neighbourhood and in the representative domain, the continuous function $g$ is nonzero on the compact sphere $\{\mathrm{\mid }u\mathrm{\mid }=\epsilon \}$, so $\mathrm{\mid }g\mathrm{\mid }$ attains a positive minimum there (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, clause 2) and $g(\epsilon v)\mathrm{\ne }0$ on $S^{n-1}$; the displayed map is then defined and smooth. Neither the chart nor $\epsilon$ is part of the value, by the two independence statements cited above.
• Convention $I-Df$, not $Df-I$. The displacement is $g(u)=u-\widehat{f}(u)$, i.e. $I-Df$ linearized at a fixed point. With the opposite ordering $\widehat{f}(u)-u$ the value is multiplied by $(-1{)}^n$, the degree of the antipodal map of $S^{n-1}$; Guillemin and Pollack use $d{f}_x-I$, so their local numbers differ from the ones on this page by $(-1{)}^n$. All items on this page use the $I-Df$ convention.
• No orientation of $M$ is used. The two spheres in the displayed map are the source and target of a single Euclidean chart expression, both oriented by the standard orientation of ${\mathbb{R}}^n$; an orientation of $M$ never enters the definition, and none is required for it.