Editing Lefschetz fixed-point theorem (section)

==Formal statement==
For a formal statement of the theorem, let

:<math>f\colon X \rightarrow X\,</math>

be a [[continuous map]] from a compact [[triangulable space]] <math>X</math> to itself. Define the '''Lefschetz number'''{{r|EncLNumbers}} <math>\Lambda_f</math> of <math>f</math> by

:<math>\Lambda_f:=\sum_{k\geq 0}(-1)^k\mathrm{tr}(H_k(f,\Q)),</math>

the alternating (finite) sum of the [[matrix trace]]s of the linear maps [[Singular homology#Functoriality|induced]] by <math>f</math> on <math>H_k(X,\Q)</math>, the [[singular homology]] groups of <math>X</math> with [[rational number|rational]] coefficients.

A simple version of the Lefschetz fixed-point theorem states: if

:<math>\Lambda_f \neq 0\,</math>

then <math>f</math> has at least one fixed point, i.e., there exists at least one <math>x</math> in <math>X</math> such that <math>f(x) = x</math>.  In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map [[homotopic]] to <math>f</math> has a fixed point as well.

Note however that the converse is not true in general: <math>\Lambda_f</math> may be zero even if <math>f</math> has fixed points, as is the case for the identity map on odd-dimensional spheres.