In mathematics, the fixed-point index is a concept in topological fixed-point theory, and in particular Nielsen theory. The fixed-point index can be thought of as a multiplicity measurement for fixed points.
The index can be easily defined in the setting of complex analysis: Let f(z) be a holomorphic mapping on the complex plane, and let z<sub>0</sub> be a fixed point of f. Then the function f(z) â z is holomorphic, and has an isolated zero at z<sub>0</sub>. We define the fixed-point index of f at z<sub>0</sub>, denoted i(f, z<sub>0</sub>), to be the multiplicity of the zero of the function f(z) â z at the point z<sub>0</sub>.
In real Euclidean space, the fixed-point index is defined as follows: If x<sub>0</sub> is an isolated fixed point of f, then let g be the function defined by
Then g has an isolated singularity at x<sub>0</sub>, and maps the boundary of some deleted neighborhood of x<sub>0</sub> to the unit sphere. We define i(f, x<sub>0</sub>) to be the Brouwer degree of the mapping induced by g on some suitably chosen small sphere around x<sub>0</sub>.
The importance of the fixed-point index is largely due to its role in the LefschetzâÂÂHopf theorem, which states:
where Fix(f) is the set of fixed points of f, and ÃÂ<sub>f</sub> is the Lefschetz number of f.
Since the quantity on the left-hand side of the above is clearly zero when f has no fixed points, the LefschetzâÂÂHopf theorem trivially implies the Lefschetz fixed-point theorem.