Nielsen fixed-point theory

Builds on Continuous map Path homotopy Homotopic maps Lefschetz number

Definition

Let $X$ be a compact connected polyhedron, that is, a space homeomorphic to a finite simplicial complex. Let $f \colon X \to X$ be a continuous map with fixed point set $\mathrm{Fix}(f) = \set{x \in X : f(x) = x}$ .

Two fixed points $x, y$ are Nielsen equivalent if there is a path $c$ from $x$ to $y$ such that $f \circ c$ is path homotopic to $c$ .
The equivalence classes are the fixed point classes of $f$ ; there are finitely many, and each is open and closed in $\mathrm{Fix}(f)$ .
Each fixed point class $F$ has a fixed-point index $i(f, F) \in \Z$ . For a single fixed point $x$ in an open set of $\R^n$ , it is the local degree of $u \mapsto u - f(u)$ at $x$ .
A class $F$ is essential if $i(f, F) \neq 0$ .

The Nielsen number $N(f)$ is the number of essential fixed point classes of $f$ .

Remarks

The sum of the indices of all fixed point classes is the Lefschetz number $L(f)$ . The number $N(f)$ depends only on the homotopy class of $f$ , and every map homotopic to $f$ has at least $N(f)$ fixed points.

Example: on the circle, let $f(z) = z^k$ with $k \neq 1$ . The fixed points are the $\abs{k - 1}$ roots of $z^{k-1} = 1$ ; they lie in distinct classes, each of index $\pm 1$ , so $N(f) = \abs{1 - k} = \abs{L(f)}$ .

Non-example of essentiality: for the identity of $S^1$ , the single fixed point class is all of $S^1$ . Its index is $L(\id) = \chi(S^1) = 0$ , so it is not essential. Indeed a small rotation has no fixed point, and $N(\id) = 0$ .

For compact manifolds of dimension at least $3$ , Wecken's theorem says that some map homotopic to $f$ has exactly $N(f)$ fixed points.

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Tag	`GEQV0DZS3W`
URI	`m2://GEQV0DZS3W`
Kind	definition
Writing	W0 stub
MSC	55M20, 54H25
Defines	Nielsen fixed-point theory
Uses	Continuous map, Path homotopy, Homotopic maps, Lefschetz number
Changed	2 November 2026

Nielsen fixed-point theory

Definition

Remarks

Reading path

Further reading

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