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(z)" be a holomorphic mapping on the complex plane, and let "z"₀ be a fixed point of "f". Then the function "f"("z") − "z" is holomorphic, and has an isolated zero at "z"₀. We define the fixed point index of "f" at "z"₀, denoted "i"("f", "z"₀), to be the multiplicity of the zero of the function "f"("z") − "z" at the point "z"₀.

In real Euclidean space, the fixed point index is defined as follows: If "x"₀ is an isolated fixed point of "f", then let "g" be the function defined by

:$g(x)\; =\; frac\{f(x)\; -\; x\}.$

Then "g" has an isolated singularity at "x"₀, and maps the boundary of some deleted neighborhood of "x"₀ to the unit sphere. We define "i"("f", "x"₀) to be the Brouwer degree of the mapping induced by "g" on some suitably chosen small sphere around "x"₀.

The Lefschetz-Hopf theorem

The importance of the fixed point index is largely due to its role in the Lefschetz-Hopf theorem, which states:

:$sum\_\{x\; in\; mathrm\{Fix\}(f)\}\; i(f,x)\; =\; Lambda\_f,$

where Fix("f") is the set of fixed points of "f", and Λ_"f" 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.