  The proof of Brouwer's fixed-point theorem based on Sperner's lemma is often
presented as an elementary combinatorial alternative to advanced proofs based
on algebraic topology. The goal of this note is to show that: (i) the
combinatorial proof of Sperner's Lemma can be considered as a cochain-level
version, written in the combinatorial language, of a standard cohomological
argument; (ii) the standard deduction of Brouwer's theorem from Sperner's lemma
is similar to the usual deduction of Brouwer's theorem from the no-retraction
theorem and is closely related to the notion of a simplicial approximation. In
order to make these connections transparent, we included the above mentioned
standard arguments, so the note is self-contained modulo some basic ideas of
combinatorial topology.
