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Suppose ''f'' : ''D''<sup> ''n''</sup> <tt>-></tt> ''D''<sup> ''n''</sup> is a continuous function that has no fixed point. The idea is to show that this leads to a contradiction. For each ''x'' in ''D''<sup> ''n''</sup>, consider the straight line that passes through ''f''(''x'') and ''x''. There is only one such line, because ''f''(''x'') ≠ ''x''. Following this line from ''f''(''x'') through ''x'' leads to a point on ''S''<sup> ''n''<tt>-</tt>1</sup>. Call this point ''g''(''x''). This gives us a continuous function ''g'' : ''D''<sup> ''n''</sup> <tt>-></tt> ''S''<sup> ''n''<tt>-</tt>1</sup>. This is a special type of continuous function known as a retraction: every point of the [[codomain]] (in this case ''S''<sup> ''n''<tt>-</tt>1</sup>) is a fixed point of the function. Intuitively it seems unlikely that there could be a retraction of ''D''<sup> ''n''</sup> onto ''S''<sup> ''n''<tt>-</tt>1</sup>, and in the case ''n'' = 1 it is obviously impossible because ''S''<sup> 0</sup> isn't even connected. For ''n'' > 1, however, proving the impossibility of the retraction is considerably more difficult. One way is to make use of [[homology group|homology groups]]: it can be shown that ''H''<sub>''n''<tt>-</tt>1</sub>(''D''<sup> ''n''</sup>) is trivial while ''H''<sub>''n''<tt>-</tt>1</sub>(''S''<sup> ''n''<tt>-</tt>1</sup>) is infinite [[cyclic group|cyclic]]. This shows that the retraction is impossible, because a retraction cannot increase the size of homology groups.
There is also an almost elementary [[combinatorial proof]]. Its main step consists in establishing [[Sperner's lemma]] in ''n'' dimensions.
== Generalizations ==
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