Editing Lefschetz fixed-point theorem (section)

==Sketch of a proof==
First, by applying the [[simplicial approximation theorem]], one shows that if <math>f</math> has no fixed points, then (possibly after subdividing <math>X</math>) <math>f</math> is homotopic to a fixed-point-free [[simplicial map]] (i.e., it sends each simplex to a different simplex).  This means that the diagonal values of the matrices of the linear maps induced on the [[Simplicial homology|simplicial chain complex]] of <math>X</math> must be all be zero.  Then one notes that, in general, the Lefschetz number can also be computed using the alternating sum of the matrix traces of the aforementioned linear maps (this is true for almost exactly the same reason that the [[Euler characteristic#Topological definition|Euler characteristic has a definition in terms of homology groups]]; see [[#Relation to the Euler characteristic|below]] for the relation to the Euler characteristic).  In the particular case of a fixed-point-free simplicial map, all of the diagonal values are zero, and thus the traces are all zero.