Sperner's Lemma, the Brouwer Fixed-Point Theorem, and Cohomology

Details Sperner's Lemma, the Brouwer Fixed-Point Theorem, and Cohomology Sperner's Lemma, the Brouwer Fixed-Point Theorem, and Cohomology

2009-06-29

arxiv.org/abs/0906.5193v1

Mathematics

Geometric Topology

10 pages

Scientific paper

The proof of the Brouwer 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 the Brouwer Theorem from Sperner's Lemma is similar to the usual deduction of the Brouwer 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 the basic notions of (simplicial) cohomology theory.

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