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.