An isolated fixed point splits under perturbation, preserving its index

Choose a target chart $(\phi ,U)$ at $x$ with $\phi (x)=0$ and restrict the representative to $W=\phi (U\cap f^{-1}(U)\cap V)$. It contains $0$. Choose $R>0$ with ${\bar{B}}_R\subset W$ and $d(u):=u-\widehat{f}(u)\mathrm{\ne }0$ on $0<\mathrm{\mid }u\mathrm{\mid }\le R$. Fix $0<r<R$ and use [F4] to choose $\rho :{\mathbb{R}}^n\to [0,1]$ equal to one near ${\bar{B}}_r$ and supported in $B_R$. The compact annulus $K=\{r\le \mathrm{\mid }u\mathrm{\mid }\le R\}$ is nonempty, and [F5] gives $m={\min }_K\mathrm{\mid }d\mathrm{\mid }>0$. The compact image $\widehat{f}({\bar{B}}_R)$ lies in the open target chart image; a finite cover by balls with doubled radii inside that image supplies $\eta >0$ such that adding a vector of norm less than $\eta$ stays in it.

By [F4] choose a regular value $a$ of $d$ with $\mathrm{\mid }a\mathrm{\mid }<\min (m\mathrm{/}2,\eta )$. Define $g_t(p)={\phi }^{-1}(\widehat{f}(u)+t\rho (u)a)$ for $u=\phi (p)\in {\bar{B}}_R$, and $g_t=f$ elsewhere, for $0\le t\le 1$. These definitions agree on an open collar of the boundary because $\mathrm{supp}\rho ⋐B_R$. They give a smooth homotopy, supported in the compact set ${\phi }^{-1}(\mathrm{supp}\rho )\subset V$, with $g_0=f$; put $g=g_1$. The vector $a$ can be arbitrarily small.

On $K$, the displacement $d-\rho a$ has norm at least $m-\mathrm{\mid }a\mathrm{\mid }>0$. Inside $B_r$ one has $\rho =1$ on a neighbourhood, so the fixed points of $g$ are exactly the preimages of the regular value $a$ under $d$. Their displacement derivative is $Dd$, which is invertible there. Hence they are nondegenerate by [F3]. The zero set is closed in ${\bar{B}}_R$ and discrete, so it is finite by [F5].

The fields $d$ and $d-\rho a$ have identical nonzero boundary values on $\mathrm{\partial }{B}_R$. By [F2] their index sums agree. The first field has only the zero $0$, of index ${\mathrm{ind}}_x(f)$ by [F1]; each zero of the second has the corresponding fixed-point index by [F3]. This proves the local sum identity. Outside the support, $g=f$ on a neighbourhood of each old fixed point, so the germ clause of The local fixed point index is independent of chart, ball and neighbourhood preserves its index. Countable choice is inherited from Sard.

For an arbitrary $V$, the fixed points of $f$ lying in it form a finite set. Choose mutually disjoint balls $B_z⋐V$ around all of them, each isolating its centre and satisfying step 1.1. Perform steps 1.1–4.1 in each ball. The supports are disjoint and the formulas equal $f$ near every ball boundary, so they glue to one smooth map and one smooth supported homotopy. No new fixed point occurs outside the balls, and every old fixed point inside $V$ was included; hence every new fixed point in $V$ is nondegenerate. Summing the local identities gives global index preservation. Since there are finitely many bumps and all perturbation vectors may be chosen arbitrarily small, any prescribed smooth-neighbourhood bound is met by taking their finitely many vectors small enough.