Lefschetz fixed-point theorem

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In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space ${\displaystyle X}$ to itself by means of traces of the induced mappings on the homology groups of ${\displaystyle X}$. It is named after Solomon Lefschetz, who first stated it in 1926.

The counting is subject to an imputed 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).