Editing Lefschetz fixed-point theorem (section)

{{Short description|Counts the fixed points of a continuous mapping from a compact topological space to itself}}
{{More footnotes needed|date=March 2022}}
In [[mathematics]], the '''Lefschetz fixed-point theorem'''<ref name="Lef1926">{{cite journal | first=Solomon|last= Lefschetz|authorlink=Solomon Lefschetz | title=Intersections and transformations of complexes and manifolds | journal=[[Transactions of the American Mathematical Society]] | year=1926 | volume=28 | pages=1–49 | doi=10.2307/1989171 | issue=1|jstor= 1989171|mr=1501331 | doi-access=free }}</ref> is a formula that counts the [[fixed point (mathematics)|fixed point]]s of a [[continuous function (topology)|continuous mapping]] from a [[compact space|compact]] [[topological space]] <math>X</math> to itself  by means of [[trace (linear algebra)|trace]]s of the induced mappings on the [[homology group]]s of <math>X</math>. It is named after [[Solomon Lefschetz]], who first stated it in 1926.

The counting is subject to an imputed [[Multiplicity (mathematics)|multiplicity]] at a fixed point called the [[fixed-point index]]. A weak version of the theorem is enough to show that a mapping without ''any'' fixed point must have rather special topological properties (like a rotation of a circle).