I am studying some very basic fixed point theory and I bumped into this interesting Wikipedia table https://en.wikipedia.org/wiki/Sperner%27s_lemma#Equivalent_results.

Here Sperner's Lemma, Brouwer Fixed Point Theorem, KMM lemma are presented as equivalent, and indeed I have read about this, moreover I know about the homotopy theory proof of Brower theorem.

What I would like to learn about is the other equivalences anf relations presented in the table, but more in general about any other results and correspondences linking discrete, combinatorial, structures (such as simplicial subdivisions) and algebraic topology. Is there some general result ? Can the fundamental group or the homology group be seen as a means to give a discrete structure to the space?

I beg pardon for having maybe formulated this question in a vague way, but I am just at the begininning. Finally any link to notes/references, especially if not too technical would be definetly appreciated.


Francesco, I would like to offer three links, which I think could be helpful for your question.

(1) This one is linking to a very interesting discussion whether Sperner's Lemma and Brouwer's FPT are really equivalent in a rigorous sense. https://mathoverflow.net/q/131413/156936

(2) This one is linking to a simple example where Tucker's Lemma directly follows from Sperner's Lemma. In other words, the table in the Wikipedia article is not so obvious, and I have the feeling that Tucker's Lemma viewed as "stronger" (because it implies Sperner's Lemma) is not a totally justified view. https://mathoverflow.net/q/362025/156936

(3) Maybe the best answer and the most thorough answer to your question is given in the excellent two articles of Nikolai Ivanov here https://arxiv.org/abs/0906.5193v2 and https://arxiv.org/abs/1909.00940

Finally, just in case you are interested in Sperner's Lemma more broadly, please take a look at this post https://mathoverflow.net/q/364040/156936

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    $\begingroup$ Thanks a lot for your very thorough answer! $\endgroup$ – Francesco Bilotta Jul 8 '20 at 8:38
  • $\begingroup$ @FrancescoBilotta Glad it is helpful. Let me know how you like the articles of Nikolai Ivanov. I imagine they cover your area of interest $\endgroup$ – Claus Jul 8 '20 at 11:34

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