Fixed-point index


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 be a holomorphic mapping on the complex plane, and let z0 be a fixed point of f. Then the function f − z is holomorphic, and has an isolated zero at z0. We define the fixed point index of f at z0, denoted i, to be the multiplicity of the zero of the function f − z at the point z0.
In real Euclidean space, the fixed-point index is defined as follows: If x0 is an isolated fixed point of f, then let g be the function defined by
Then g has an isolated singularity at x0, and maps the boundary of some deleted neighborhood of x0 to the unit sphere. We define i to be the Brouwer degree of the mapping induced by g on some suitably chosen small sphere around x0.

The Lefschetz–Hopf theorem

The importance of the fixed-point index is largely due to its role in the Lefschetz–Hopf theorem, which states:
where Fix is the set of fixed points of f, and Λf 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 Lefschetz–Hopf theorem trivially implies the Lefschetz fixed point theorem.