I wonder if Brouwer's fixed-point theorem and Borsuk-Ulam's theorem are equivalent.

Brouwer's fixed-point theorem (simple form). Let $B_{\mathbb{R}^{n}}[0,1]=\{x\in \mathbb{R}^n: \|x-0\|\leq 1\}$ Then, any continuous $f:B_{\mathbb{R}^{n}}[0,1]\to B_{\mathbb{R}^{n}}[0,1] $ has a fixed point, i.e. there is a $x\in B_{\mathbb{R}^{n}}[0,1]$ such that $f(x)=x$. \


Borsuk-Ulam's theorem. Let a continuous function $f:\mathbb{S}^n\to \mathbb{R}^n$ (where $\mathbb{S}^n=\partial B_{\mathbb{R}^{n+1}}[0,1]$ is the $n$-sphere). Then there is a point $x\in \mathbb{S}^n$ for which $f(x)=f(−x)$.

In other words, I would like to know if there are any proofs of the implications below:

  • Brouwer's fixed-point theorem implies Borsuk-Ulam's theorem.

  • Borsuk-Ulam's theorem implies Brouwer's fixed-point theorem.

I would be happy if there were a simple (or as self-contained as possible) proof that could be reproduced here. But as the proofs of these theorems (at least in the bibliographic research I have done) are intricate depending on various definitions and lemas I think such simple proofs do not exist. In that case I would be satisfied with a set of references (books or articles) that had proof of confirmation or proof of the negative of both of the implications above.

My bibliographic research is summarized as follows:

The Wikipedia entry on Borsuk-Ulam's theorem states that such theorems are equivalent ( see here ) but provide no direct proof. And if I understood correctly, it is not possible to conclude by any chain of implications on other theorems set forth therein. Also the references of Wikipedia do not bring any proof of this affirmation.

  • $\begingroup$ The more common notation for what you write as $B_{\mathbb{R}^n}[0, 1]$ is $B^n$. $\endgroup$ – anomaly Nov 3 '17 at 23:54
  • 1
    $\begingroup$ For starters, the following article gives an elementary proof of the implication B-U $\implies$ Brouwer. $\endgroup$ – Bass Nov 4 '17 at 1:17

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