Fixed point index

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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 z₀ be a fixed point of f. Then the function f(z) − z is holomorphic, and has an isolated zero at z₀. We define the fixed point index of f at z₀, denoted i(f, z₀), to be the multiplicity of the zero of the function f(z) − 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 zero at x₀, and maps some deleted neighborhood of x₀ to the unit sphere. We define i(f, x₀) to be the Brouwer degree of the mapping induced by g on some suitably chosen small sphere around x₀.

[edit] 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(f) is the set of fixed points of f, and L(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.