 Fixed-point index

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; thenThe 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 $i(f,a)=\mathrm{sign}\det (I-Df(a))$. 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.