Lefschetz fixed-point theorem

In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact triangulable topological space

X

{\displaystyle X}

to itself by means of traces of the induced mappings on the homology groups of

X

{\displaystyle X}

. It is named after Solomon Lefschetz, who first stated it in 1926 but in different way involving coincidence points of functions.

There are different versions of this theorem: the weak version of theorem shows only existence of fixed point when expression dependent on traces for a mapping is nonzero. The stronger version of theorem, sometimes called Lefschetz-Hopf theorem counts fixed points with respect to their fixed-point index, provided that their number is finite. There is also algebraic geometry counterpart of this theorem called Lefschetz trace formula that allows to express number of points of variety over finite field in terms of action of Frobenius morphism on its cohomologies.

The topological versions of Lefschetz fixed-point theorem are generalizations of other classical results in topology like Brouwer fixed-point theorem or Poincare-Hopf theorem.

Historical context

Lefschetz presented his fixed-point theorem in his 1926 paper about mappings on manifolds. Lefschetz's focus was not on fixed points of maps, but rather on what are now called coincidence points of maps.

Lefschetz defined coincidence number for two functions as an alternating sum of traces of maps induced on homologies and cohomologies by two functions and isomorphisms arising from Poincare duality for both manifolds. He proved that if this number is nonzero, then

f

{\displaystyle f}

and

g

{\displaystyle g}

must have a coincidence point.