{title2=$i(f,a)$}

= Fixed point indices
{synonym}

The local fixed point index assigns an <integer> to an isolated <fixed point> $a$ of a <continuous map> $f:U\to\mathbb R^n$, where $U$ is a neighbourhood of $a$. Choose a small ball $B$ containing no other <fixed point>; then
$$
i(f,a)=\deg(x\mapsto x-f(x),B,0).
$$
The <mapping degree> counts the zero of the displacement map with its local topological multiplicity. Changing the ball or deforming $f$ without allowing a fixed point onto its boundary leaves this index unchanged.

If $f$ is differentiable and $I-Df(a)$ is invertible, the first-order displacement gives $i(f,a)=\operatorname{sign}\det(I-Df(a))$. On the real line, $f(x)=x/2$ has index $+1$ at zero, while $f(x)=2x$ has index $-1$ there. A nonzero index forces the existence of a <fixed point> in the region. Under the hypotheses of the <Lefschetz-Hopf fixed-point theorem>, the sum of local indices is the global <Lefschetz number>.


 Fixed point index (source code)