Editing Brouwer fixed-point theorem (section)

===A proof using the Lefschetz fixed-point theorem===
The Lefschetz fixed-point theorem says that if a continuous map ''f'' from a finite simplicial complex ''B'' to itself has only isolated fixed points, then the number of fixed points counted with multiplicities (which may be negative) is equal to the Lefschetz number
:<math>\displaystyle \sum_n(-1)^n\operatorname{Tr}(f|H_n(B))</math>
and in particular if the Lefschetz number is nonzero then ''f'' must have a fixed point. If ''B'' is a ball (or more generally is contractible) then the Lefschetz number is one because the only non-zero [[simplicial homology]] group is: <math>H_0(B)</math> and ''f'' acts as the identity on this group, so ''f'' has a fixed point.<ref>{{harvnb|Hilton|Wylie|1960}}</ref><ref>{{harvnb|Spanier|1966}}</ref>