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# The Fixpoint Theorem of Brouwer

Norbert A'Campo

The following proof of the Fixpoint Theorem of Brouwer was outlined as classroom exercise at the University of Montpellier in the early sixties.

Theorem 1 (Brouwer)   Let be the unit Euclidean ball in and let be a continuous mapping. Then there exists with .

a. Reduction to the -differentiable case. Let be the function . Let given by: on and equal to in the complement of , where the constant is chosen such that the integral of over equals . Let be continuous. Let be the extension of to , for which we have on . For put . Show that the restriction of to maps into . Show that the mappings are continuously differentiable and approximate in the topology of uniform convergence the mapping . Show that if there exists a continuous mapping without fixpoints, then there will also exist a continuously differential map without fixpoints. It follows, that it suffices to proof the Brouwer Fixpoint Theorem only for continuously differentiable mappings.

b. Proof for -differentiable mappings. The proof is by contradiction. Assume, that the continuously differential mapping has no fixpoints. Let the mapping, such that for every point the points are in that order on a line of . The mapping is also continuously differentiable and satisfies for . We write and have for the identities: . Observe since holds. The following is a contradiction:

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