Brouwer fixed-point theorem
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

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: H 0 (B) {\displaystyle H_{0}(B)} and f acts as the identity on this group, so f has a fixed point.
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