Editing Brouwer fixed-point theorem (section)

===A proof using homology or cohomology===
The proof uses the observation that the [[boundary (topology)|boundary]] of the ''n''-disk ''D''<sup>''n''</sup> is ''S''<sup>''n''−1</sup>, the (''n'' − 1)-[[sphere]].

[[Image:Brouwer fixed point theorem retraction.svg|thumb|right|Illustration of the retraction ''F'']]
Suppose, for contradiction, that a continuous function {{nowrap|''f'' : ''D''<sup>''n''</sup> → ''D''<sup>''n''</sup>}} has ''no'' fixed point. This means that, for every point x in ''D''<sup>''n''</sup>, the points ''x'' and ''f''(''x'') are distinct. Because they are distinct, for every point x in ''D''<sup>''n''</sup>, we can construct a unique ray from ''f''(''x'') to ''x'' and follow the ray until it intersects the boundary ''S''<sup>''n''−1</sup> (see illustration). By calling this intersection point ''F''(''x''), we define a function ''F''&nbsp;:&nbsp;''D''<sup>''n''</sup>&nbsp;→&nbsp;''S''<sup>''n''−1</sup> sending each point in the disk to its corresponding intersection point on the boundary.  As a special case, whenever ''x'' itself is on the boundary, then the intersection point ''F''(''x'') must be ''x''.

Consequently, ''F'' is a special type of continuous function known as a [[retraction (topology)|retraction]]: every point of the [[codomain]] (in this case ''S''<sup>''n''−1</sup>) is a fixed point of ''F''.

Intuitively it seems unlikely that there could be a retraction of ''D''<sup>''n''</sup> onto ''S''<sup>''n''−1</sup>, and in the case ''n'' = 1, the impossibility is more basic, because ''S''<sup>0</sup> (i.e., the endpoints of the closed interval ''D''<sup>1</sup>) is not even connected. The case ''n'' = 2 is less obvious, but can be proven by using basic arguments involving the [[fundamental group]]s of the respective spaces: the retraction would induce a surjective [[group homomorphism]] from the fundamental group of ''D''<sup>2</sup> to that of ''S''<sup>1</sup>, but the latter group is isomorphic to '''Z''' while the first group is trivial, so this is impossible. The case ''n'' = 2 can also be proven by contradiction based on a theorem about non-vanishing [[vector field]]s.

For ''n'' > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of [[Homology (mathematics)|homology groups]]:  the homology ''H''<sub>''n''−1</sub>(''D''<sup>''n''</sup>) is trivial, while ''H''<sub>''n''−1</sub>(''S''<sup>''n''−1</sup>) is infinite [[cyclic group|cyclic]]. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group.

The impossibility of a retraction can also be shown using the [[de Rham cohomology]] of open subsets of Euclidean space ''E''<sup>''n''</sup>. For ''n'' ≥ 2, the de Rham cohomology of ''U'' = ''E''<sup>''n''</sup> – (0) is one-dimensional in degree 0 and ''n'' – 1, and vanishes otherwise. If a retraction existed, then ''U'' would have to be contractible and its de Rham cohomology in degree ''n'' – 1 would have to vanish, a contradiction.<ref>{{harvnb|Madsen|Tornehave |1997|pages=39–48}}</ref>