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Then Brouwer's fixed-point theorem is one of the most celebrated theorems of analysis and topology that emerged in the first half of the twentieth century. Since the second half of the twentieth century various proofs of this theorem have arisen. Much of this proofs seeks simplicity and clarity. See for exemple

In these excellent notes of Mathematical Analysis written by Professor Bruce Blackadar there exists in pages $1217$, $1218$ and $1219$ a proof scheme of Brouwer's fixed-point theorem. Although incomplete, they seem promising to me.

The proof scheme is basically the following: First, it is proved that the three theorems below are equivalent. Second, it proves that the last theorem is true.

Theorem. [ Brouwer Fixed-Point Theorem}] Every continuous function $f : B^n \to B^n$ has a fixed point, i.e. there is a point $x \in B^n$ with $f (x) = x$. \

Theorem. [No-Retraction Theorem] There is no retraction from $B^n$ to $S^{n-1}$ \

Theorem. [No-Contraction Theorem] The sphere $S^{n−1}$ is not contractible, i.e. the identity map on $S^{n−1}$ is not homotopic to a constant map.

And the point is precisely this, the proof scheme of Brouwer's fixed-point theorem seems to me to be incompletely written because it seems to me very simple or very short writing. See pages $1217$, $1218$ and $1219$ in Bruce Blackadar's notes.

My question is just this one. How to complete this idea of the proof of the fixed point theorem of Brouwer of rigorous way? Can this proof scheme be found in any reference other than Professor Bruce Blackadar's notes?

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    $\begingroup$ At page 143 and beyond of my notes you can find a constructive and very simple approach to Brouwer's theorem through Sperner's Lemma. $\endgroup$ – Jack D'Aurizio Nov 11 '17 at 13:43
  • $\begingroup$ @JackD'Aurizio, Very interesting your notes. They look very well written to me. Although the proof of Brouwer's theorem through Sperner's lemma is restricted to dimension 2, it seems to me that it generalizes naturally to any dimension. But I'll look at yours more carefully as soon as possible. $\endgroup$ – MathOverview Nov 11 '17 at 13:53
  • $\begingroup$ I appreciate you like them. You are right, in order to simplify the exposition some results are not stated in full generality, but I make an explicit mention of the fact that, like Brouwer's theorem for the unit disk in $\mathbb{R}^2$ can be proved through a colorful combinatorial argument, the same holds for the Borsuk-Ulam theorem (through Tucker's and Ki-Fan's lemmas in simplicial geometry). $\endgroup$ – Jack D'Aurizio Nov 11 '17 at 13:57
  • $\begingroup$ +1 for @JackD'Aurizio comment. The use of Sperner's lemma is a gem and I have not seen any easier proof of Brouwer Fixed Point Theorem. $\endgroup$ – Paramanand Singh Nov 11 '17 at 17:22

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