Brouwer fixed-point theorem: Difference between revisions

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===A proof by Hirsch===

There is also a quick proof, by [[Morris Hirsch]], based on the impossibility of a differentiable retraction. Let ''f'' denote a continuous map from the unit ball Dⁿ in n-dimensional Euclidean space to itself and assume that ''f'' fixes no point. By continuity and the fact that Dⁿ is compact, it follows that for some ε > 0, ∥x - ''f''(x)∥ > ε for all x in Dⁿ. Then the map ''f'' can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the [[Weierstrass approximation theorem]] or by [[convolution|convolving]] with smooth [[bump function]]s. One then defines a retraction as above whichby sending each x to the point of ∂Dⁿ where the unique ray from x through ''f(x)'' intersects ∂Dⁿ, and this must now be a differentiable mapping. Such a retraction must have a non-singular value p ∈ ∂Dⁿ, by [[Sard's theorem]], which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse image ''f''^-1(p) would be a 1-manifold with boundary. Its boundary would have to contain at least two endpoints, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction.<ref>{{harvnb|Hirsch|1988}}</ref>

R. Bruce Kellogg, Tien-Yien Li, and [[James A. Yorke]] turned Hirsch's proof into a [[Computability|computable]] proof by observing that the retract is in fact defined everywhere except at the fixed points.{{sfn|Kellogg|Li|Yorke|1976}} For almost any point, ''q'', on the boundary, (assuming it is not a fixed point) the one manifold with boundary mentioned above does exist and the only possibility is that it leads from ''q'' to a fixed point. It is an easy numerical task to follow such a path from ''q'' to the fixed point so the method is essentially computable.{{sfn|Chow|Mallet-Paret|Yorke|1978}} gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.